Algebra I System of Linear Equations
|
|
- Ezra Black
- 5 years ago
- Views:
Transcription
1 1
2 Algebra I System of Linear Equations
3 Table of Contents Click on the topic to go to that section Solving Systems by Graphing Solving Systems by Substitution Solving Systems by Elimination Choosing your Strategy Writing Systems to Model Situations Teacher Note Standards 3
4 Solving Systems by Graphing Return to Table of Contents 4
5 System of Equations A system of linear equations is comprised of 2 or more linear equations. The solution of the system will be the values of the variables which make all the equations true. 5
6 Solve by Graphing Example: y = 2x - 4 The graph of the line that represents the solutions to the above equation is shown y It represents all the points whose x and y values make the above equation true x The line is easy to find from the equation since the equation is in slopeintercept form
7 Solve by Graphing Example: y y = x Similarly, this graph is of the line that represents the solutions to this equation. It represents all the points whose x and y values make the above equation true x -10 7
8 Solve by Graphing Example: y = 2x - 4 y = x y Here are the lines that represent the solutions to both those equations. Each line shows the infinite set of solutions for each equation. What must be true about the point at which they cross? x DISCUSS. 8
9 Solve by Graphing Example: y = 2x - 4 y = x y At the point they cross, both equations must be true, since that point is on both lines x They appear to cross at (3, 2). Let's check that in both equations
10 Solve by Graphing Example: Substitute x = 3 and y = 2 into both equations and see if both equations are true. y = 2x y (2) = 2(3) = 2 correct x y = x + 5 (2) = -(3) = 2 correct 10
11 System of Equations Not all systems have solutions...and some have an infinite number of solutions. Let's see how to figure out whether there are solutions, how many, and what they are. Click here to watch a music video that introduces what we will learn about systems. 11
12 The Number of Solutions When graphing two lines there are three possibilities. They meet in one point: the point of intersection. They never meet: they are parallel. They meet at all their points: they are the same line. 12
13 The Number of Solutions So, systems of equations can have either: 1 solution, if the lines meet at one point 0 solutions, if they never meet Infinite solutions, if they are the same line 13
14 Type 1: One Solution The two lines intersect in exactly ONE place. y 10 The solution is the point at which they intersect. 5 The slopes of the lines must be different, or they would never cross x
15 Type 1: One Solution y y = 2x - 4 y = x This is the example we started with. 5 As we confirmed there is one solution to this system of equations: (3, 2) x
16 Type 2: No Solution The lines never meet. There is no solution true for both lines. The lines are parallel y They must have the same slope, since they are parallel x But, they must have different intercepts, or they would be the same line
17 Type 2: No Solution y = 2x + 6 y = 2x + 2 Both are written in slope intercept form y 10 y = mx + b to make it easy to compare slopes and y-intercepts. The slope for both lines is 2 (the coefficient of x). So, the lines are parallel x The y-intercepts are different, +6 and +2, so the lines never cross
18 Type 3: Infinite Solutions The lines overlap at all points. y 10 They are different equations for the same line. 5 The lines are parallel. So, they must have the same slopes x The intercepts are the same, since all their points are the same
19 Type 3: Infinite Solutions y = 2x + 2 y = 2x + 2 y Both are written in slope intercept form 10 y = mx + b to make it easy to compare slopes and y- intercepts. The slope for both lines is 2 (the coefficient of x). So, the lines are parallel x The y-intercepts for both lines are +2, so the lines overlap everywhere
20 Type 3: Infinite Solutions y = 2x + 2 y = 2x + 2 In slope intercept form, the fact that these are the same line is obvious. y 10 But, if the equations were written as below, it would be less obvious: 5 2y - 4x = 4-6x = -3y x That's why it's always a good idea to put equations into slope-intercept form...they're easier to read, graph and compare
21 The Number of Solutions First, put the equations into slope-intercept form by solving for y. Then, decide on the number of solutions. After that, solutions can be found in three different ways. 21
22 How can you quickly decide the number of solutions a system has? 1 Solution No Solution Math Practice Infinitely Many 22
23 Solving both Equations for y Let's solve this system of equations y = -5x x + 2y = 6 The equation on the left is in slope-intercept form. Do you see that the slope is -5 and its y-intercept is +4? The equation on the right is not in slope-intercept form, so we can't see it's slope or y-intercept. So, we can't tell yet how many solutions will satisfy both equations. Let's solve the second equation for y. 23
24 Solving for y 10x + 2y = 6-10x -10x Subtract 10x from both sides 2y = -10x + 6 Divide both sides by 2 2y = -10x y = -5x + 3 This is now in slope-intercept form. We can see the slope and y-intercept m = -5 b = 3 24
25 Solving both Equations for y y= -5x + 4 Original Equations 10x + 2y = 6 y= -5x + 4 Slope Intercept Form y = -5x + 3 m = -5 b = 4 Slopes and Intercepts m = -5 b = 3 The slopes are the same but the y-intercepts are different. How many solutions are there? 25
26 Solving both Equations for y Let's solve this system of equations y = 2x + 5 6x + 2y = 4 The equation on the left is in slope-intercept form and we can see the slope is +2 and the y-intercept is +5. The equation on the right is not in slope-intercept form, let's solve that equation for y. 26
27 Solving for y 6x + 2y = 4-6x -6x Subtract 6x from both sides 2y = -6x + 4 Divide both sides by 2 2y = -6x y = -3x + 2 This is now in slope-intercept form. m = -3 b = +2 27
28 Solving both Equations for y y= -5x + 4 Original Equations 6x + 2y = 4 y= -5x + 4 Slope Intercept Form y = -3x + 2 m = -5 b = 4 Slopes and Intercepts m = -3 b = +2 The slopes are different. How many solutions are there? 28
29 1 How many solutions does this system have: y = 2x - 7 y = 3x + 8 A 1 solution Answer B C no solution infinitely many solutions 29
30 2 How many solutions does this system have: 3x - y = -2 y = 3x + 2 Answer A B C 1 solution no solution infinitely many solutions 30
31 3 How many solutions does this system have: 3x + 3y = 8 1 y = x 3 Answer A B C 1 solution no solution infinitely many solutions 31
32 4 How many solutions does this system have: y = 4x 2x - 0.5y = 0 Answer A B C 1 solution no solution infinitely many solutions 32
33 5 How many solutions does this system have: 3x + y = 5 6x + 2y = 1 Answer A B C 1 solution no solution infinitely many solutions 33
34 Consider this... Suppose you are walking to school. Your friend is 5 blocks ahead of you. You can walk two blocks per minute and your friend can walk one block per minute. How many minutes will it take for you to catch up with your friend? Math Practice 34
35 Solution First, make a table to represent the problem. Time (min.) Friend's distance from your start (blocks) Your distance from your start (blocks) 35
36 Solution Continued Next, plot the points on a graph. Blocks Time (min.) Friend's distance from your start (blocks) Your distance from your start (blocks) Time (min.) 36
37 Solution Continued The point where the lines intersect is the solution to the system. Blocks (5, 10) is the solution In the context of this problem this means after 5 minutes, you will meet your friend at block Time (min.) 37
38 Example Solve this system of equations graphically: y = 2x - 3 y = x y Answer x
39 Example Solve the system of equations graphically: y = -3x + 4 y y = x Answer x
40 Checking Your Work Given the graph below, what is the point of intersection? y = -3x - 1 y = 4x y x
41 Checking Your Work Now take the ordered pair we just found and substitute it into the equations to prove that it is a solution for BOTH lines. (-1, 2) y = -3x - 1 (2) = -3(-1) = = 2 y = 4x + 6 (2) = 4(-1) = = 2 Math Practice 41
42 6 Solve the following system by graphing: y y = -x + 4 y = 2x + 1 A (3, 1) 10 5 Answer B (1, 3) Click for answer choices AFTER students have graphed the system C (-1, 3) x D (1, -3)
43 7 Solve the following system by graphing: 1 y = x y = x 1 2 A (0,-1) 10 5 y Answer B (0,0) Click for answer choices AFTER students have graphed the system C (-1, 0) x D (0, 1)
44 8 Solve the following system by graphing: y = x y = x A (0, 4) 10 5 y Answer B (-4, 2) Click for answer choices AFTER students have graphed the system C (5, 6) x D (2, 5)
45 Graphing Quickly Transforming linear equations into slope-intercept form usually saves time in the end. It also makes it easy to check your work. 45
46 Example Solve the following system of linear equations by graphing: 2x + y = 5 -x + y = 2 Step 1: Rewrite the linear equations in slope-intercept form 2x + y = 5-2x -2x y = -2x + 5 -x + y = 2 +x +x y = x
47 Solution Continued Step 2: Plot the y-intercept and use the slope to plot the second point y = -2x + 5 y-intercept = (0, 5) slope = -2 slope= (down 2, right 1) 10 5 y y = x x y-intercept = (0, 2) -5 slope = 1 slope= (up 1, right 1)
48 Solution Continued Step 3: Locate the point of intersection and check your work: (1, 3) y y = -2x + 5 (3) = -2(1) = = y = x x (3) = (1) =
49 Example Solve the system of equations graphically: 2x + y = 3 x - 2y = 4 Step 1: Rewrite in slope-intercept form 2x + y = 3-2x -2x y = -2x + 3 x - 2y = 4 -x -x -2y = -x + 4-2y = -x y = x
50 Solution Continued Step 2: Plot y-intercept and use slope to plot second point y-intercept = (0, 3) y slope = slope= (down 2, right 1) 5 y-intercept = (0, -2) 1 slope = 2 slope= (up 1, right 2) x
51 Solution Continued Step 3: Locate the Point of Intersection and check your work: (2, -1) y y = -2x + 3 (-1) = -2(2) = = y = x ( 1) = (2) = x 1 =
52 9 What is the solution of the system of linear equations provided on the graph? A (0, 1) B (1, 0) 10 5 y Answer C (2, 3) D (3, 2) x
53 10 Which graph below represents the solution to the following system of linear equations: -x + 2y = 2 3y = x + 6 Answer A C B D 53
54 11 Solve the following system by graphing: x 3y = 3 y = x 7 10 y A (3, 4) B (9, 2) Click for answer choices AFTER students have C graphed infintely the system many D no solution x Answer
55 Example Solve the system of equations graphically: y = 3x + 6 9x - 3y = -18 Step 1: Rewrite in slope-intercept form y = 3x + 6 9x - 3y = -18-9x -9x -3y = -9x -18-3y = -9x y = 3x
56 Solution Continued Step 2: Plot y-intercept and use slope to plot second point y = 3x + 6 y-intercept = (0, 6) slope = 3 slope= (up 3, right 1) y = 3x + 6 y-intercept = (0, 6) slope = 3 slope= (up 3, right 1) 56
57 Solution Continued Step 3: Locate the Point of Intersection and check your work: infinite amount of points: infinite solutions y = 3x + 6 9x - 3y =
58 Example Solve the system of equations graphically: 4x - 2y = 10 8x - 4y = 12 Step 1: Rewrite in slope-intercept form 4x - 2y = 10-4x -4x -2y = -4x y = -4x y = 2x - 5 8x - 4y = 12-8x 8x -4y = -8x y = -8x y = 2x
59 Solution Continued Step 2: Plot y-intercept and use slope to plot second point y = 2x - 5 y-intercept = (0, -5) slope = 2 slope= (up 2, right 1) y = 2x -3 y-intercept = (0, -3) slope = 2 slope= (up 2, right 1) 59
60 Solution Continued Step 3: Locate the Point of Intersection and check your work: no point of intersection: no solution y = 2x - 5 y = 2x
61 12 Solve the this system by graphing: y = 3x + 4 y 4y = 12x Answer 5 A (2, 4) B (0.4, 2.2) Click for answer choices AFTER students have graphed the system C D infinitely many solutions no solution x 61
62 13 Solve the this system by graphing: y = 3x + 4 4y = 12x y Answer A (3,4) 5 B (-3,-4) C Click for answer choices AFTER students have graphed the system infinitely many x D no solution
63 Solving Systems by Substitution Return to Table of Contents 63
64 Example Solve the system of equations graphically. y = x y y = -2x Teacher Note Why was it difficult to solve Click this for Additional system by graphing? Question x
65 Substitution Explanation Graphing can be inefficient or approximate. Another way to solve a system of linear equations is to use substitution. Substitution allows you to create a one variable equation. 65
66 Solving by Substitution Step 1: If you are not given a variable already alone, find the EASIEST variable to solve for (get it alone) Step 2: Substitute the expression into the other equation and solve for the variable Step 3: Substitute the numerical value you found into EITHER equation and solve for the other variable. Write the solution as (x, y) 66
67 Example Solve the system using substitution: y = x y = -2x Step 1: Choose an equation from the system and substitute it into the other equation y = x First Equation y = -2x Second Equation x = -2x Substitute First Equation into Second Equation 67
68 Step 2: Solve the new equation x = -2x x +2x Add 2x to both sides 3x = Subtract 6.1 from both sides 3x = Solution Continued 3x = Divide both sides by 3 3 x = -2.5 This is the value of x for our solution...now we have to find y. 68
69 Solution Continued Step 3: Substitute the solution x = -2.5 into either equation and solve. y = x y = (-2.5) y = 3.6 y = -2x y = -2(-2.5) y = y = 3.6 The solution to the system of linear equations is (-2.5, 3.6). We only had to plug the x value into one of the equations to get this. The second one just provides a check. If it comes out the same, our solution must be correct. 69
70 Good Practice After you evaluate the solution, it is good practice is to double check your work by substituting the solution into both equations. CHECK: See if (-2.5, 3.6) satisfies both equations y = -2x (3.6) = -2(-2.5) = = 3.6 y = x (3.6) = (-2.5) = 3.6 Math Practice If your checks end in true statements, the solution is correct. 70
71 Example Solve the system using substitution: 2x - 3y = -1 y = x - 1 Step 1: Substitute one equation into the other equation. Since one equation is already solved for y, I'll substitute that into the other equation. 2x - 3y = -1 y = x - 1 2x - 3(x - 1) = -1 71
72 Solution Continued Step 2: Solve the new equation 2x - 3(x - 1) = -1 2x - 3x + 3 = -1 x = 4 Step 3: Substitute the solution into either equation and solve 2x - 3y = -1 2(4) - 3y = y = -1-3y = -9 y = 3 (4, 3) You end with the correct answer with either equation you use for this step. y = x - 1 y = (4) - 1 y = 3 (4, 3) 72
73 Example Continued Check: See if (4, 3) satisfies both equations 2x - 3y = -1 2(4) - 3(3) = = -1-1 = -1 y = x - 1 (3) = (4) = 3 Math Practice The ordered pair satisfies both equations so the solution is (4, 3) 73
74 14 Solve by substitution: y = x - 3 y = -x + 5 A (4, 9) B (-4, -9) Click for answer choices AFTER students have solved the system C (4, 1) Answer D (1, 4) 74
75 15 Solve by substitution: y = x y = -3x 7 A (2, -8) Answer B (-3, 2) Click for answer choices AFTER students have C infinitely solved the system many solutions D no solutions 75
76 16 Solve by substitution. y = 4x x + 3y = -1 A (4, 5) B (5, 4) Click for answer choices AFTER students have C solved infintely the system many solutions D no solutions Answer 76
77 17 Solve by substitution. y = 8x x + 3y = 0 A (-2, -2) B (-2, 2) Click for answer choices AFTER students have solved the system C (2, -2) Answer D (2, 2) 77
78 18 Solve by substitution. 8x + 3y = -9 y = 3x + 14 A (-8, 5) B (7, 5) Click for answer choices AFTER students have solved the system C (-3, 5) Answer D (-7, 5) 78
79 Choosing a Variable Sometimes, a variable is not already isolated. When this occurs, you need to determine which one you would prefer to isolate and finish solving the system with the substitution method. Examine each system of equations. Which variable would you choose to substitute? y = 4x y = -2x + 9 Why? -y + 4x = -1 x - 4y = 1 2x + 4y = -10-8x - 3y = -12 Teacher Note 79
80 19 Examine this system of equations. Which variable could quickly be solved for and substituted into the other equation? y = -2x + 5 2y = 10-4x Answer A B x y 80
81 20 Examine this system of equations. Which variable could quickly be solved for and substituted into the other equation? 2y - 8 = x y + 2x = 4 A x Answer B y 81
82 21 Examine this system of equations. Which variable could quickly be solved for and substituted into the other equation? x - y = 20 2x + 3y = 0 Answer A B x y 82
83 Rewriting Sometimes you need to rewrite one of the equations so that you can use the substitution method. For example: The system: 3x - y = 5 2x + 5y = -8 Solve for y: 3x - y = 5-3x 3x -y = -3x Which letter is the easiest to solve for? The "y" in the first equation because Click to discuss which letter there is only a "-1" as the coefficient. So, the original system is equivalent to: y = 3x - 5 Click to see 2x + 5y = -8 y = 3x
84 Solution Continued Now Substitute and Solve: y = 3x - 5 2x + 5 y = -8 2x + 5(3x - 5) = -8 2x + 15x - 25 = -8 17x - 25 = -8 17x = 17 x = 1 84
85 Solution Continued Substitute x = 1 into one of the equations. 2(1) + 5y = y = -8 5y = -10 y = -2 The ordered pair (1, -2) satisfies both equations in system. Math Practice 3x - y = 5 3(1) - (-2) = = 5 5 = 5 2x + 5y = -8 2(1) + 5(-2) = = -8-8 = -8 85
86 22 Solve using substitution. 6x + y = 6-3x + 2y = -18 A (-6, 2) B (6, -2) Click for answer choices AFTER students have C solved (-6 the, -2) system Answer D (2, -6) 86
87 23 Solve using substitution. 2x - 8y = 20 -x + 6y = -12 A (6, -1) Answer B (-6, 5) Click for answer choices AFTER students have solved the system C (5, 5) D (-6, -1) 87
88 24 Solve using substitution. -3x - 3y = 12-4x - 7y = 7 Answer A (-3, -7) B (-7, 3) Click for answer choices AFTER students have solved the system C (3, 7) D (7, 3) 88
89 25 Solve the system by substitution: y = x - 6 y = -4 Answer A (-10, -4) B Click (-4, for 2) answer choices AFTER students have C (2, -4) solved the system D (10, 4) 89
90 26 Solve the system by substitution: y + 2x = -14 y = 2x + 18 A (1, 20) B Click (1, for 18) answer choices AFTER C students (8, -2) have solved the D (-8, system 2) Answer 90
91 Example Solve this system using substitution: x + y = 6 5x + 5y = 10 x + y = 6 x = 6 - y 5(6 - y) + 5y = y + 5y = = 10 - solve the first equation for x - substitute 6 - y for x in 2nd equation - solve for y - This is FALSE! Since 30 = 10 is a false statement, the system has no solution. Answer: NO SOLUTION 91
92 Example Solve the following system using substitution: x + 4y = -3 2x + 8y = -6 x + 4y = -3 - solve the first equation for x x = -3-4y 2(-3-4y) + 8y = -6 - sub. -3-4y for x in 2nd equation -6-8y + 8y = -6 - solve for y -6 = -6 - This is ALWAYS TRUE! Since -6 = -6 is always a true statement, there are infinitely many solutions to the system. Answer: Infinite Solutions 92
93 27 Solve the system by substitution: 4x = -5y + 50 x = 2y - 7 A (6, 6.5) Answer B Click (5, for 6) answer choices AFTER C students (4, 5) have solved the D (6, system 5) 93
94 28 Solve the system by substitution: y = -3x y + 4x = 19 A (6, 5) Answer B Click (-7, for 5) answer choices AFTER C students (42, -103) have solved the D (6, system -5) 94
95 29 Solve the system using substitution. 3 4 x + y = 2 6x + 8y = 16 Answer A (-4, 5) B (4, Click -1) for answer choices AFTER students Chave infinitely solved many the solutions system D no solutions 95
96 30 Solve using substitution. 16x + 2y = -5 y = -8x - 6 A (-3, -1) Answer B No Solution Click for answer Cchoices Infinite AFTER Solutions students have D (-1, solved -3) the system 96
97 Solving System by Elimination Return to Table of Contents 97
98 Standard Form Recall that the Standard Form of a linear equation is: Ax + By = C When both linear equations of a system are in standard form the system can be solved by using elimination. The elimination strategy adds or subtracts the equations in the system to eliminate a variable. 98
99 Additive Inverses Let's talk about what's happening with these numbers = 3 + (-3) = -5x + 5x = 9x + (-9x) = 99
100 Choosing a Variable How do you decide which variable to eliminate? First: Look to see if one variable has the same or opposite coefficients. If so, eliminate that variable. Math Practice 100
101 Addition or Subtraction If the variables have the same coefficient, subtract the two equations to eliminate the variable. Same Coefficients { 3x 3x Subtract { 3x -(3x) 0x If the variables have opposite coefficients, add the two equations to eliminate the variable. Opposite Coefficients { 3x -3x Add { 3x + -3x) 0x 101
102 Example Solve the following system by elimination: 5x + y = 44-4x - y = -34 Step 1: Choose which variable to eliminate The y in both equations have opposite coefficients so they will be the easiest to eliminate Step 2: Add the two equations 5x + y = 44-4x - y = -34 x + 0y = 10 x =
103 Solution Continued Step 3: Substitute the solution into either equation and solve x = 10 5(10) + y = y = 44 y = -6 The solution to the system is (10, -6) Math Practice Check: 5x + y = 44 5(10) + (-6) = = = 44-4x - y = -34-4(10) - (-6) = = =
104 Example Solve the following system by elimination: 3x + y = 15-3x - 3y = -21 Step 1: Choose which variable to eliminate Math Practice The x in both equations have opposite coefficients so they will be the easiest to eliminate Step 2: Add the two equations 3x + y = 15-3x - 3y = -21-2y = -6 y = 3 104
105 Solution Continued Step 3: Substitute the solution into either equation and solve y = 3 3x + 3 = 15 3x = 12 x = 4 The solution to the system is (4, 3) Math Practice Check: 3x + y = 15 3(4) + 3 = = = 15-3x - 3y = -21-3(4) - 3(3) = = =
106 31 Solve the system by elimination: x + y = 6 x - y = 4 A (5, 1) Answer B Click (-5, for -1) answer choices AFTER C students (1, 5) have solved the D no system solution 106
107 32 Solve the system by elimination: 2x + y = -5 2x - y = -3 A (-2,1) Answer B Click (-1,-2) for answer choices AFTER C students (-2,-1) have solved the D infinitely systemmany 107
108 33 Solve using elimination. -2x - 8y = 10 2x - 6y = 18 A (-2, 3) B Click (4, -6) for answer choices AFTER C students (-6, 4) have solved the D (3, system -2) Answer 108
109 Multiple Methods There are 2 ways to complete the problem below using elimination. 5x + y = 17-2x + y = -4 Step 1: Choose which variable to eliminate The y in both equations have the same coefficient so they will be the easiest to eliminate Step 2: Add or Subtract the two equations First Method: Multiply one equation by -1 then add equations Second Method: Subtract equations keeping in mind that all signs change 109
110 Solution Continued First Method -1(-2x + y = -4) = 2x - y = 4 5x + y = 17 2x - y = 4 7x = 21 x = 3 Second Method 5x + y = 17 -(-2x + y = -4) 7x = 21 x = 3 Answer Why do both methods produce the same solution? 110
111 Solution Continued Step 3: Substitute the solution into either equation and solve x = 3-2(3) + y = y = -4 y = 2 The solution to the system is (3, 2) Math Practice Check: 5x + y = 17 5(3) + 2 = = = 17-2x + y = -4-2(3) + 2 = = -4-4 =
112 34 Solve the system by elimination: 2x + y = -6 A (-4, 2) 3x + y = -10 Answer B (3, 5) Click for answer choices AFTER C (4, 2) students have solved the D infinitely system many 112
113 35 Solve the system by elimination: A (2, -7) 3x + 6y = 48-5x + 6y = 32 Answer B Click (2, for 7) answer choices AFTER C students (7, 2) have solved the D infinitely system many 113
114 Common Coefficient Sometimes, it is not possible to eliminate a variable by simply adding or subtracting the equations. When this is the case, you need to multiply one or both equations by a nonzero number in order to create a common coefficient before adding or subtracting the equations. 114
115 Example Solve the following system using elimination: 3x + 4y = -10 5x - 2y = 18 The y would be the easiest variable to eliminate because 4 is a common coefficient. Multiply second equation by 2 so the coefficients are opposites. 2(5x - 2y = 18) The y coefficients are opposites, so solve by adding the equations 3x + 4y = x - 4y = 36 13x = 26 x = 2 115
116 Example Continued Solve for y, by substituting x = 2 into one of the equations. 3x + 4y = -10 3(2) + 4y = y = -10 4y = -16 y = -4 (2, -4) is the solution Check: 3x + 4y = -10 3(2) + 4(-4) = = = -10 5x - 2y = 18 5(2) - 2(-4) = = =
117 Choosing Variable to Eliminate In the previous example, the y was eliminated by finding a common coefficient of 4. Creating a common coefficient of 4 required one additional step: Multiplying the second equation by 2 3x + 4y = -10 5x - 2y = 18 Either variable can be eliminated when solving a system of equations as long as a common coefficient is utilized. 117
118 Example Solve the same system by eliminating x. 3x + 4y = -10 5x - 2y = 18 Multiply the first equation by 5 and the second equation by 3 so the coefficients will be the same 5(3x + 4y = -10) 15x + 20y = -50 3(5x - 2y = 18) 15x - 6y = 54 Now solve by subtracting the equations. 15x + 20y = (15x - 6y = 54) 26y = -104 y =
119 Example Continued Solve for x, by substituting y = -4 into one of the equations. (2, -4) is the solution. Check: 3x + 4y = -10 3x + 4(-4) = -10 3x = -10 3x = 6 x = 2 Math Practice 3x + 4y = -10 3(2) + 4(-4) = = = -10 5x - 2y = 18 5(2) - 2(-4) = = =
120 System of Equations Examine each system of equations. Which variable would you choose to eliminate? What do you need to multiply each equation by? 2x + 5y = -1 x + 2y = 0 3x + 8y = 81 5x - 6y = -39 Teacher Note 3x + 6y = 6 2x - 3y = 4 120
121 36 Which variable can you eliminate with the least amount of work in the system below? A x B y 2x + 5y = 20 3x - 10y = 37 Answer 121
122 37 Solve the following system of equations using elimination: 2x + 5y = 20 A (1, 57) 3x - 10y = 37 Answer B (1, Click 77) for answer choices AFTER 2 C students (11, - have ) solved the system 5 D infinitely many solutions 122
123 38 Which variable can you eliminate with the least amount of work in the system below? A x B y x + 3y = 4 3x + 4y = 2 Answer 123
124 39 What will you multiply the first equation by in order to solve this system using elimination? x + 3y = 4 3x + 4y = 2 Answer 124
125 40 Solve the following system of equations: x + 3y = 4 3x + 4y = 2 Answer 2 A (-2, ) 3 Click for answer choices B (-2, 1) AFTER students C (-2, 2) have solved the system D infinitely many solutions 125
126 Example Solve the following system using elimination: 9x - 5y = 4-18x + 10y = 10 The y would be the easiest variable to eliminate because 10 is a common coefficient. Multiply first equation by 2 so the coefficients are opposites. 2(9x - 5y = 4) The y coefficients are opposites, so solve by adding the equations 18x - 10y = x + 10y = 10 0 = 18 is this true? False, NO Move SOLUTION for solution 126
127 Example Solve the following system using elimination: -4x - 10y = -22 2x + 5y = 11 The x would be the easiest variable to eliminate because 4 is a common coefficient. Multiply second equation by 2 so the coefficients are opposites. 2(2x + 5y= 11 ) The y coefficients are opposites, so solve by adding the equations -4x - 10y = x +10y = 22 0 = 0 is this true? True, Move INFINITE for solution SOLUTIONS 127
128 41 Solve the system by elimination: x - y = 5 x - y = -7 A (11, -4) B Click (4, 11) for answer choices AFTER students C (-4, -11) have solved the system Answer D no solution 128
129 42 Solve using elimination. -20x - 18y = x + 9y = 14 A (-8, -1) B infinite Click for solutions answer choices AFTER C no solution students have solved the system D (-1, 8) Answer 129
130 43 Solve using elimination. 9x + 3y = 27 18x + 6y = 30 A infinite solutions B (4, Click 7) for answer choices AFTER students C (-7, 4) have solved the system Answer D no solution 130
131 Choose Your Strategy Return to Table of Contents 131
132 Choosing Strategy Systems of linear equations can be solved using any of the three methods we previously discussed. Before solving a system, an analysis of the equations should be done to determine the "best" strategy to utilize. Graphing Substitution Elimination 132
133 Example Altogether 292 tickets were sold for a basketball game. An adult ticket cost $3 and a student ticket cost $1. Ticket sales for the event were $470. How many adult tickets were sold? How many student tickets were sold? 133
134 Example Continued Step 1: Define your variables Let a = number of adult tickets Let s = number of student tickets Step 2: Set up the system number of tickets sold: a + s = 292 money collected: 3a + s =
135 Example Continued Step 3: Solve the system a + s = 292 -( 3a + s = 470 ) -2a+ 0 = -178 a = 89 Elimination was utilized for this example because the x had a common coefficient. Note 135
136 Example Continued a = 89 a + s = s = 292 s = 203 There were 89 adult tickets and 203 student tickets sold Math Practice Check: a + s = = = 292 3a + s = 470 3(89) = = =
137 44 What method would require the least amount of work to solve the following system: y = 3x - 1 y = 4x Answer A B C graphing substitution elimination 137
138 45 Solve the following system of linear equations using the method of your choice: y = 3x - 1 y = 4x A (-4, -1) B Click (-1, -4) for answer choices AFTER students C (-1, 4) have solved the system Answer D (1, 4) 138
139 46 What method would require the least amount of work to solve the following system: 4s - 3t = 8 t = -2s -1 Answer A B C graphing substitution elimination 139
140 47 Solve the following system of linear equations using the method of your choice: 4s - 3t = 8 t = -2s -1 Answer 1 1 A (-2, Click ) C 2 for answer choices (, 2) 2 AFTER students 1 have solved system B (, -2) D (2, -2) 2 140
141 48 What method would require the least amount of work to solve the following system: A B C graphing substitution elimination 3m - 4n = 1 3m - 2n = -1 Answer 141
142 49 Solve the following system of linear equations using the method of your choice: 3m - 4n = 1 3m - 2n = -1 A (-2, -1) Click for answer choices B (-1, -1) AFTER students have solved the system C (-1, 1) Answer D (1, 1) 142
143 50 What method would require the least amount of work to solve the following system: A B C graphing substitution elimination y = -2x 1 y = x Answer 143
144 51 Solve the following system of linear equations using the method of your choice: A (-6, 12) y = -x 1 y = x Answer B Click for answer choices (2, -4) AFTER students C (-2, have 2) solved the system D (1, -2) 144
145 52 What method would require the least amount of work to solve the following system: A graphing u = 4v 3u - 3v = 7 Answer B substitution C elimination 145
146 53 Solve the following system of linear equations using the method of your choice: u = 4v 3u - 3v = 7 Answer A 28 7 (, ) C (28, 7) Click 9 9for answer choices AFTER ( 7, 28 ) (7, 7 B students have solved D system)
147 54 A piece of glass with an initial temperature of 99 F is cooled at a rate of 3.5 F/min. At the same time, a piece of copper with an initial temperature of 0 F is heated at a rate of 2.5 F/min. Let m = the number of minutes and t = the temperature in F. Which system models the given scenario? Answer A B C t = m t = m t = m t = m t = m t = 0-2.5m 147
148 55 Which method would you use to solve the system from the previous question? t = Click m to Reveal t = System m Answer A B C graphing substitution elimination 148
149 56 Solve the following system of linear equations: t = Click m to Reveal t = m System Answer A m = 1 t = 2.5 B m = 1 t = 95.5 Click for C m answer = 16.5 choices t = 6.6 AFTER D m = 16.5 t = students have solved system 149
150 57 Choose a strategy and then answer the question. What is the value of the y-coordinate of the solution to the system of equations x 2y = 1 and x + 4y = 7? A 1 B -1 C 3 D 4 Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,
151 Writing Systems to Model Situations Return to Table of Contents 151
152 Creating and Solving Systems Step 1: Define the variables Step 2: Analyze components and create equations Step 3: Solve the system utilizing the best strategy 152
153 Example A group of 148 peole is spending five days at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1,410 pounds of food was ordered. Part A: Write an equation or a system of equations that describe the above situation and define your variables. a = number of adults c = number of children a + c = a + 9c = 1,410 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,
154 Example Continued Part B: Using your work from part A, find (1) the total number of adults in the group (2) the total number of children in the group a + c = a + 9c = 1,410 (1) (2) c = -a a + 9(-a + 148) = a - 9a = a = 78 a = 26 a + c = c = 148 c =
155 Example Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel ordered two slices of pizza and three colas. Tanisha s bill was $6.00, and Rachel s bill was $5.25. What was the price of one slice of pizza? What was the price of one cola? p = cost of pizza slice c = cost of cola 3p + 2c = p + 3c = 5.25 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,
156 Example Continued 3p + 2c = p + 3c = 5.25 Elimination: Multiply first equation by 2 Multiply second equation by -3 6p + 4c = 12-6p - 9c = c = c = 0.75 Cola: $0.75 Pizza: $1.50 3p + 2c = p + 2(0.75) = 6 3p = 6 3p = 4.5 p =
157 Example Your class of 22 is going on a trip. There are four drivers and two types of vehicles, vans and cars. The vans seat six people, and the cars seat four people, including drivers. How many vans and cars does the class need? Let v = the number of vans and c = the number of cars 157
158 Set Up the System Drivers: v + c = 4 People: 6v + 4c = 22 Solve the system by substitution: 158
159 Substitute, Solve and Check v + c = 4 6v + 4c = 22 solve for v substitute v = -c + 4 6(-c + 4) + 4c = 22-6c c = 22-2c + 24 = 22-2c = -2 substitute v = -(1) + 4 c = 1 v = 3 then check: c = 1; v = 3 (3) + (1) = 4 6(3) + 4(1) = 22 4 = 4 22 =
160 58 Your class receives $1,105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9. Set up a system and solve. How many packages of cards were sold? Answer You will answer how many packages of gift wrap in the next question. 160
161 59 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9. Set up a system and solve. How many packages of gift wrap were sold? Answer 161
162 60 The sum of two numbers is 47, and their difference is 15. What is the larger number? A 16 B 31 Answer C 32 D
163 61 Ramon rented a sprayer and a generator. On his first job, he used each piece of equipment for 6 hours at a total cost of $90. On his second job, he used the sprayer for 4 hours and the generator for 8 hours at a total cost of $100 What was the hourly cost for the sprayer? Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,
164 62 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many quarters do you have? Answer 164
165 63 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many nickels do you have? Answer 165
166 64 Julia went to the movies and bought one jumbo popcorn and two chocolate chip cookies for $5.00. Marvin went to the same movie and bought one jumbo popcorn and four chocolate chip cookies for $6.00. How much does one chocolate chip cookie cost? A $0.50 B $0.75 Answer C $1.00 D $2.00 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,
167 65 Mary and Amy had a total of 20 yards of material from which to make costumes. Mary used three times more material to make her costume than Amy used, and 2 yards of material was not used. How many yards of material did Amy use for her costume? Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,
168 66 The tickets for a dance recital cost $5.00 for adults and $2.00 for children. If the total number of tickets sold was 295 and the total amount collected was $1220, how many adult tickets were sold? Answer From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June,
169 67 In a basketball game, Marlene made 16 field goals. Each of the field goals were worth either 2 points or 3 points, and Marlene scored a total of 39 points from field goals. Part A Let x represent the number of two-point field goals and y represent the number of three-point field goals. Write a system of equations in terms of x and y to model the situation. When you finish, writing your answer, type the number "1" into your Responder. Answer PARCC - EOY - Question #16 Calculator Section - SMART Response Format 169
170 68 In a basketball game, Marlene made 16 field goals. Each of the field goals were worth either 2 points or 3 points, and Marlene scored a total of 39 points from field goals. Part B How many three-point field goals did Marlene make in the game? Answer PARCC - EOY - Question #16 Calculator Section 170
171 Applications: Systems of Equations We have one more Application that is an extension of Systems of Equations. It can also be found in the Dynamics Unit of our Algebra Based Physics course. Let's get started. 171
172 Extension: Tension Force The tension in a rope is the same everywhere in the rope. If two masses hang down from either side of a cable, for instance, the tension in both sides must be the same. FT1 = FT2 20 kg 50 kg "Atwood Machine" 172
173 Extension: Tension Force A 20 kg mass hangs from one end of a rope that passes over a small frictionless pulley. A 50 kg weight is suspended from the other end of the rope. Which way will the 20 kg mass accelerate? Which way will the 50 kg mass accelerate? a) Draw a Free Body Diagram for each mass b) Write the Net Force equation for each mass c) Find the equations for the tension force FT d) Find the equation for acceleration e) Find the value of the acceleration f) Find the value of the tension force 20 kg F T1 = F T2 50 kg "Atwood Machine" 173
174 Extension: Tension Force a) Draw a Free Body Diagram for each mass FT FT a 20 kg m1g F T1 = F T2 = F T 20 kg 50 kg m2g 50 kg a Answer Remember the tension in the rope is the same everywhere, so FT is the same for both masses. The direction of acceleration is also different. What about the magnitude of acceleration? 174
175 Extension: Tension Force b) Write the Net Force equation for each mass Remember there is no special equation for tension. We need to use net force to find the tension. Below each diagram, write the Net Force equation for each mass: + FT FT - Answer a - 20 kg m1g m2g 50 kg a + 175
176 - Extension: Tension Force b) Write the Net Force equation for each mass + FT F T a 20 kg - m1g 50 kg a ΣF = m1a FT - m1g = m1a m 2 g + Answer ΣF = m 2 a -F T + m 2 g = m 2 a What do you notice about how the signs were chosen for the various forces? 176
177 Extension: Tension Force c) Find the equations for the tension force F T We have two equations (one for each mass) and two unknowns (F T and a). This means we can combine the equations together to solve for each variable! F T - m 1 g = m 1 a -F T + m 2 g = m 2 a Solve each for F T : F T - m 1 g = m 1 a F T = m 1 g + m 1 a -F T + m 2 g = m 2 a F T = m 2 g - m 2 a Now we can set them equal to one another: m 1 g + m 1 a = m 2 g - m 2 a 177
178 Extension: Tension Force d) Find the equation for the acceleration Now we can combine the tension equations m 1 g + m 1 a = F T There is only one unknown (a) here. Solve for a: Add m 2 a and subtract m 1 g from both sides: factor out 'a' : (remember factoring is just the opposite of distributing) divide by (m 1 + m 2 ): F T = m 2 g - m 2 a m 1 g + m 1 a = m 2 g - m 2 a m 1 a + m 2 a = m 2 g - m 1 g a(m 1 + m 2 ) = m 2 g - m 1 g a = m 2 g - m 1 g m 1 + m 2 178
179 Extension: Tension Force e) Find the value of the acceleration Substitute and solve: Remember: this is the acceleration for both m1 and m2. 20kg 50kg Answer 179
180 Extension: Tension Force f) Find the value of the tension Now we can use either equation to solve for Tension: Answer 20kg 50kg 180
181 Standards Return to Table of Contents 181
182 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Math Practice 182
Algebra I. Systems of Linear Equations and Inequalities. Slide 1 / 179. Slide 2 / 179. Slide 3 / 179. Table of Contents
Slide 1 / 179 Algebra I Slide 2 / 179 Systems of Linear Equations and Inequalities 2015-04-23 www.njctl.org Table of Contents Slide 3 / 179 Click on the topic to go to that section 8th Grade Review of
More informationAlgebra I. Slide 1 / 176 Slide 2 / 176. Slide 3 / 176. Slide 4 / 176. Slide 6 / 176. Slide 5 / 176. System of Linear Equations.
Slide 1 / 176 Slide 2 / 176 Algebra I Sstem of Linear Equations 21-11-2 www.njctl.org Slide 3 / 176 Slide 4 / 176 Table of Contents Solving Sstems b Graphing Solving Sstems b Substitution Solving Sstems
More informationAlgebra I. Systems of Linear Equations and Inequalities. 8th Grade Review. Slide 1 / 179 Slide 2 / 179. Slide 4 / 179. Slide 3 / 179.
Slide 1 / 179 Slide 2 / 179 lgebra I Systems of Linear Equations and Inequalities 2015-04-23 www.njctl.org Slide 3 / 179 Table of Contents Click on the topic to go to that section 8th Grade Review of Systems
More informationSystems of Equations and Inequalities
1 Systems of Equations and Inequalities 2015 03 24 2 Table of Contents Solving Systems by Graphing Solving Systems by Substitution Solve Systems by Elimination Choosing your Strategy Solving Systems of
More informationSimple Inequalities Involving Addition and Subtraction. Unit 3 Inequalities.notebook. November 18, Table of Contents
Table of Contents Simple Inequalities Addition/Subtraction Simple Inequalities Multiplication/Division Two-Step and Multiple-Step Inequalities Solving Compound Inequalities Special Cases of Compound Inequalities
More information6th Grade. Translating to Equations. Slide 1 / 65 Slide 2 / 65. Slide 4 / 65. Slide 3 / 65. Slide 5 / 65. Slide 6 / 65
Slide 1 / 65 Slide 2 / 65 6th Grade Dependent & Independent Variables 15-11-25 www.njctl.org Slide 3 / 65 Slide 4 / 65 Table of Contents Translating to Equations Dependent and Independent Variables Equations
More informationAlgebra I Solving & Graphing Inequalities
Slide 1 / 182 Slide 2 / 182 Algebra I Solving & Graphing Inequalities 2016-01-11 www.njctl.org Slide 3 / 182 Table of Contents Simple Inequalities Addition/Subtraction click on the topic to go to that
More informationAlgebra I. Slide 1 / 79. Slide 2 / 79. Slide 3 / 79. Equations. Table of Contents Click on a topic to go to that section
Slide 1 / 79 Slide 2 / 79 Algebra I Equations 2015-08-21 www.njctl.org Table of Contents Click on a topic to go to that section. Slide 3 / 79 Equations with the Same Variable on Both Sides Solving Literal
More informationSlide 2 / 79. Algebra I. Equations
Slide 1 / 79 Slide 2 / 79 Algebra I Equations 2015-08-21 www.njctl.org Slide 3 / 79 Table of Contents Click on a topic to go to that section. Equations with the Same Variable on Both Sides Solving Literal
More informationFoundations of Math. Chapter 3 Packet. Table of Contents
Foundations of Math Chapter 3 Packet Name: Table of Contents Notes #43 Solving Systems by Graphing Pg. 1-4 Notes #44 Solving Systems by Substitution Pg. 5-6 Notes #45 Solving by Graphing & Substitution
More information6th Grade. Equations & Inequalities.
1 6th Grade Equations & Inequalities 2015 12 01 www.njctl.org 2 Table of Contents Equations and Identities Tables Determining Solutions of Equations Solving an Equation for a Variable Click on a topic
More informationUnit 5 SIMULTANEOUS LINEAR EQUATIONS
MATH 8 Unit 5 SIMULTANEOUS LINEAR EQUATIONS By the end of this unit, students should be able to: 1. Solve simultaneous linear equations by graphing. 2. Understand what it means to solve a system of equations.
More information8th Grade. Equations with Roots and Radicals.
1 8th Grade Equations with Roots and Radicals 2015 12 17 www.njctl.org 2 Table of Contents Radical Expressions Containing Variables Click on topic to go to that section. Simplifying Non Perfect Square
More informationLesson 12: Systems of Linear Equations
Our final lesson involves the study of systems of linear equations. In this lesson, we examine the relationship between two distinct linear equations. Specifically, we are looking for the point where the
More information8th Grade The Number System and Mathematical Operations Part
Slide 1 / 157 Slide 2 / 157 8th Grade The Number System and Mathematical Operations Part 2 2015-11-20 www.njctl.org Slide 3 / 157 Table of Contents Squares of Numbers Greater than 20 Simplifying Perfect
More informationName Class Date. What is the solution to the system? Solve by graphing. Check. x + y = 4. You have a second point (4, 0), which is the x-intercept.
6-1 Reteaching Graphing is useful for solving a system of equations. Graph both equations and look for a point of intersection, which is the solution of that system. If there is no point of intersection,
More information8th Grade. The Number System and Mathematical Operations Part 2.
1 8th Grade The Number System and Mathematical Operations Part 2 2015 11 20 www.njctl.org 2 Table of Contents Squares of Numbers Greater than 20 Simplifying Perfect Square Radical Expressions Approximating
More informationSample: Do Not Reproduce LF6 STUDENT PAGES LINEAR FUNCTIONS STUDENT PACKET 6: SYSTEMS OF LINEAR EQUATIONS. Name Period Date
Name Period Date LINEAR FUNCTIONS STUDENT PACKET 6: SYSTEMS OF LINEAR EQUATIONS LF6.1 LF6.2 LF6.3 Introduction to Systems of Linear Equations Understand the definition of a system of linear equations Understand
More informationGrade 8 Systems of Linear Equations 8.EE.8a-c
THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 8 Systems of Linear Equations 8.EE.8a-c 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES THE NEWARK
More informationAlgebra I. Simple Inequalities Involving Addition and Subtraction. Slide 1 / 182 Slide 2 / 182. Slide 4 / 182. Slide 3 / 182.
Slide 1 / 182 Slide 2 / 182 lgebra I Solving & Graphing Inequalities 2016-011 www.njctl.org Slide 3 / 182 Slide 4 / 182 Table of ontents Simple Inequalities ddition/subtraction click on the topic to go
More informationAlgebra I. Slide 1 / 182. Slide 2 / 182. Slide 3 / 182. Solving & Graphing Inequalities. Table of Contents
Slide 1 / 182 Slide 2 / 182 lgebra I Solving & Graphing Inequalities 2016-01-11 www.njctl.org Table of ontents Slide 3 / 182 Simple Inequalities ddition/subtraction click on the topic to go to that section
More information6th Grade. Dependent & Independent Variables
Slide 1 / 68 Slide 2 / 68 6th Grade Dependent & Independent Variables 2014-10-28 www.njctl.org Slide 3 / 68 Table of Contents Translating to Equations Dependent and Independent Variables Click on a topic
More informationHow can you use linear functions of two independent variables to represent problem situations?
Problems that occur in business situations often require expressing income as a linear function of one variable like time worked or number of sales. For example, if an employee earns $7.25 per hour, then
More informationWarm Up. Unit #1: Basics of Algebra
1) Write an equation of the given points ( 3, 4) & (5, 6) Warm Up 2) Which of the following choices is the Associative Property 1) 4(x + 2) = 4x + 8 2) 4 + 5 = 5 + 4 3) 5 + ( 5) = 0 4) 4 + (3 + 1) = (4
More information8th Grade The Number System and Mathematical Operations Part
Slide 1 / 157 Slide 2 / 157 8th Grade The Number System and Mathematical Operations Part 2 2015-11-20 www.njctl.org Slide 3 / 157 Table of Contents Squares of Numbers Greater than 20 Simplifying Perfect
More informationUnit 12: Systems of Equations
Section 12.1: Systems of Linear Equations Section 12.2: The Substitution Method Section 12.3: The Addition (Elimination) Method Section 12.4: Applications KEY TERMS AND CONCEPTS Look for the following
More informationEquations with the Same Variable on Both Sides
Equations with the Same Variable on Both Sides Previously, you solved equations with variables on one side, similar to the following: Now, we will be given an equation with the same variable on both sides.
More informationSlide 1 / 178 Slide 2 / 178. Click on a topic to go to that section.
Slide / 78 Slide 2 / 78 Algebra I The Number System & Mathematical Operations 205--02 www.njctl.org Slide 3 / 78 Slide 4 / 78 Table of Contents Review of Natural Numbers, Whole Numbers, Integers and Rational
More informationAlgebra I. Slide 1 / 178. Slide 2 / 178. Slide 3 / 178. The Number System & Mathematical Operations. Table of Contents
Slide 1 / 178 Slide 2 / 178 Algebra I The Number System & Mathematical Operations 2015-11-02 www.njctl.org Table of Contents Slide 3 / 178 Review of Natural Numbers, Whole Numbers, Integers and Rational
More information1. What are the various types of information you can be given to graph a line? 2. What is slope? How is it determined?
Graphing Linear Equations Chapter Questions 1. What are the various types of information you can be given to graph a line? 2. What is slope? How is it determined? 3. Why do we need to be careful about
More informationNew Jersey Center for Teaching and Learning. Progressive Mathematics Initiative
Slide 1 / 70 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students
More informationHIGLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL ALIGNMENT
HIGLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL ALIGNMENT 8 th Grade Math Third Quarter Unit 3: Equations Topic C: Equations in Two Variables and Their Graphs (Continued) This topic focuses on extending the
More informationAlgebra I. Slide 1 / 178. Slide 2 / 178. Slide 3 / 178. The Number System & Mathematical Operations. Table of Contents
Slide 1 / 178 Slide 2 / 178 Algebra I The Number System & Mathematical Operations 2015-11-02 www.njctl.org Table of Contents Slide 3 / 178 Review of Natural Numbers, Whole Numbers, Integers and Rational
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II 1 st Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 1 st Nine Weeks, 2016-2017 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationName: Essential Skills Practice for students entering Geometry or Accelerated Geometry
Name: Essential Skills Practice for students entering Geometry or Accelerated Geometry Use this document to review the mathematics that you have learned previously. Completion of the Essential Skills Practice
More informationOTHER METHODS FOR SOLVING SYSTEMS
Topic 18: Other methods for solving systems 175 OTHER METHODS FOR SOLVING SYSTEMS Lesson 18.1 The substitution method 18.1 OPENER 1. Evaluate ab + 2c when a = 2, b = 3, and c = 5. 2. Following is a set
More informationChapter 9 Solving Systems of Linear Equations Algebraically
Name: Chapter 9 Solving Systems of Linear Equations Algebraically 9.1 Solving Systems of Linear Equations by Substitution Outcomes: 1. Interpret algebraic reasoning through the study of relations 9. Solve
More informationCommon Core Algebra Rock the Regents Station 1:Linear Equations & Inequalities. Name: Teacher: Date: Grade: (circle one) Period:
Common Core Algebra Rock the Regents 2016 Station 1:Linear Equations & Inequalities Name: Teacher: Date: Grade: 9 10 11 12 (circle one) Period: Topic: Modeling Expressions Tips/Hints Look for keywords/hints
More informationSystems of Linear Equations: Solving by Adding
8.2 Systems of Linear Equations: Solving by Adding 8.2 OBJECTIVES 1. Solve systems using the addition method 2. Solve applications of systems of equations The graphical method of solving equations, shown
More informationWRITING EQUATIONS through 6.1.3
WRITING EQUATIONS 6.1.1 through 6.1.3 An equation is a mathematical sentence that conveys information to the reader. It uses variables and operation symbols (like +, -, /, =) to represent relationships
More informationAlgebra I. Slide 1 / 79. Slide 2 / 79. Slide 3 / 79. Equations. Table of Contents Click on a topic to go to that section
Slide 1 / 79 Slide 2 / 79 lgebra I Equations 2015-08-21 www.njctl.org Table of ontents lick on a topic to go to that section. Slide 3 / 79 Equations with the Same Variable on oth Sides Solving Literal
More informationAlgebra I Practice Exam
Algebra I This practice assessment represents selected TEKS student expectations for each reporting category. These questions do not represent all the student expectations eligible for assessment. Copyright
More informationUnit 6 Systems of Equations
1 Unit 6 Systems of Equations General Outcome: Develop algebraic and graphical reasoning through the study of relations Specific Outcomes: 6.1 Solve problems that involve systems of linear equations in
More informationUnit 4 Systems of Linear Equations
Number of Days: 23 1/29/18 3/2/18 Unit Goals Stage 1 Unit Description: Students extend their knowledge of linear equations to solve systems of linear equations. Students solve systems of linear equations
More informationName Period Date Ch. 5 Systems of Linear Equations Review Guide
Reteaching 5-1 Solving Systems by Graphing ** A system of equations is a set of two or more equations that have the same variables. ** The solution of a system is an ordered pair that satisfies all equations
More informationGrade 6 The Number System & Mathematical Operations
Slide 1 / 206 Slide 2 / 206 Grade 6 The Number System & Mathematical Operations 2015-10-20 www.njctl.org Slide 3 / 206 Table of Contents Addition, Natural Numbers & Whole Numbers Addition, Subtraction
More informationStrategic Math. General Review of Algebra I. With Answers. By: Shirly Boots
Strategic Math General Review of Algebra I With Answers By: Shirly Boots 1/6 Add/Subtract/Multiply/Divide Addmoves to the right -3-2 -1 0 1 2 3 Subtract moves to the left Ex: -2 + 8 = 6 Ex: -2 8 = - 10
More informationChapter 1-2 Add and Subtract Integers
Chapter 1-2 Add and Subtract Integers Absolute Value of a number is its distance from zero on the number line. 5 = 5 and 5 = 5 Adding Numbers with the Same Sign: Add the absolute values and use the sign
More informationInequalities Chapter Test
Inequalities Chapter Test Part 1: For questions 1-9, circle the answer that best answers the question. 1. Which graph best represents the solution of 8 4x < 4 A. B. C. D. 2. Which of the following inequalities
More informationCCGPS Coordinate Algebra. EOCT Review Units 1 and 2
CCGPS Coordinate Algebra EOCT Review Units 1 and 2 Unit 1: Relationships Among Quantities Key Ideas Unit Conversions A quantity is a an exact amount or measurement. A quantity can be exact or approximate
More informationConsistent and Dependent
Graphing a System of Equations System of Equations: Consists of two equations. The solution to the system is an ordered pair that satisfies both equations. There are three methods to solving a system;
More information7th Grade Math. Expressions & Equations. Table of Contents. 1 Vocab Word. Slide 1 / 301. Slide 2 / 301. Slide 4 / 301. Slide 3 / 301.
Slide 1 / 301 Slide 2 / 301 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial
More informationFOR STUDENTS WHO HAVE COMPLETED ALGEBRA 1 (Students entering Geometry)
FOR STUDENTS WHO HAVE COMPLETED ALGEBRA (Students entering Geometry) Dear Parent/Guardian and Student, Name: Date: Period: Attached you will find a review packet of skills which each student is expected
More informationLesson 28: Another Computational Method of Solving a Linear System
Lesson 28: Another Computational Method of Solving a Linear System Student Outcomes Students learn the elimination method for solving a system of linear equations. Students use properties of rational numbers
More informationTopic 1. Solving Equations and Inequalities 1. Solve the following equation
Topic 1. Solving Equations and Inequalities 1. Solve the following equation Algebraically 2( x 3) = 12 Graphically 2( x 3) = 12 2. Solve the following equations algebraically a. 5w 15 2w = 2(w 5) b. 1
More information3.3 Solving Systems with Elimination
3.3 Solving Systems with Elimination Sometimes it is easier to eliminate a variable entirely from a system of equations rather than use the substitution method. We do this by adding opposite coefficients
More informationAlgebra I The Number System & Mathematical Operations
Slide 1 / 178 Slide 2 / 178 Algebra I The Number System & Mathematical Operations 2015-11-02 www.njctl.org Slide 3 / 178 Table of Contents Review of Natural Numbers, Whole Numbers, Integers and Rational
More informationMath 1 Unit 7 Review
Name: ate: 1. Which ordered pair is the solution to this system of equations? 5. system of equations is graphed on the set of axes below. y = x + 4 x + y = 2. (1, 5). (0, 2). ( 1, 3). ( 4, 0) 2. Which
More informationGrade 6. The Number System & Mathematical Operations.
1 Grade 6 The Number System & Mathematical Operations 2015 10 20 www.njctl.org 2 Table of Contents Addition, Natural Numbers & Whole Numbers Addition, Subtraction and Integers Multiplication, Division
More informationDue for this week. Slide 2. Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
MTH 209 Week 1 Due for this week Homework 1 (on MyMathLab via the Materials Link) Monday night at 6pm. Read Chapter 6.1-6.4, 7.1-7.4,10.1-10.3,10.6 Do the MyMathLab Self-Check for week 1. Learning team
More informationAlgebra I The Number System & Mathematical Operations
Slide 1 / 72 Slide 2 / 72 Algebra I The Number System & Mathematical Operations 2015-08-21 www.njctl.org Slide 3 / 72 Table of Contents Review of Natural Numbers, Whole Numbers & Integers Review of Rational
More information8 th Grade Domain 2: Algebra and Functions (40%) Sara
8 th Grade Domain 2: Algebra and Functions (40%) 1. Tara creates a budget for her weekly expenses. The graph shows how much money is in the account at different times. Find the slope of the line and tell
More informationFoundations of Algebra. Learning Goal 3.1 Algebraic Expressions. a. Identify the: Variables: Coefficients:
Learning Goal 3.1 Algebraic Expressions What you need to know & be able to do 1. Identifying Parts of Algebraic Expressions 3.1 Test Things to remember Identify Parts of an expression Variable Constant
More informationGrade 8. Functions 8.F.1-3. Student Pages
THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 8 Functions 8.F.1-3 Student Pages 2012 2012 COMMON CORE CORE STATE STATE STANDARDS ALIGNED ALIGNED MODULES Grade 8 - Lesson 1 Introductory Task
More informationName. Check with teacher. equation: a. Can you find. a. (-2, -3) b. (1, 3) c. (2, 5) d. (-2, -6) a. (-2, 6) b. (-1, 1) c. (1, 3) d. (0, 0) Explain why
7.1 Solving Systems of Equations: Graphing Name Part I - Warm Up with ONE EQUATION: a. Which of the following is a solution to the equation: y 3x 1? a. (-2, -3) b. (1, 3) c. (2, 5) d. (-2, -6) Partt II
More informationChapter Systems of Equations
SM-1 Name: 011314 Date: Hour: Chapter 6.1-6.4 Systems of Equations 6.1- Solving Systems by Graphing CCS A.REI.6: Solve systems of equations exactly and approximately (e.g. with graphs), focusing on pairs
More informationKeystone Exam Concept Review. Properties and Order of Operations. Linear Equations and Inequalities Solve the equations. 1)
Keystone Exam Concept Review Name: Properties and Order of Operations COMMUTATIVE Property of: Addition ASSOCIATIVE Property of: Addition ( ) ( ) IDENTITY Property of Addition ZERO PRODUCT PROPERTY Let
More informationCHAPTER 4: SIMULTANEOUS LINEAR EQUATIONS (3 WEEKS)...
Table of Contents CHAPTER 4: SIMULTANEOUS LINEAR EQUATIONS (3 WEEKS)... 2 4.0 ANCHOR PROBLEM: CHICKENS AND PIGS... 6 SECTION 4.1: UNDERSTAND SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS... 8 4.1a Class Activity:
More informationNAME DATE PER. Review #11 Solving Systems of Equations 1. Write the linear function that includes the points (4, 9) and (-2, -6).
Review #11 NAME DATE PER. Review #11 Solving Systems of Equations 1. Write the linear function that includes the points (4, 9) and (-2, -6). 2. The graph of a line that contains the points (-3, 2) and
More informationALGEBRA 1 UNIT 3 WORKBOOK CHAPTER 6
ALGEBRA 1 UNIT 3 WORKBOOK CHAPTER 6 FALL 2014 0 1 Algebra 1 Section 6.1 Notes: Graphing Systems of Equations System of Equations: a set of two or more equations with the same variables, graphed in the
More information5th Grade. Slide 1 / 191. Slide 2 / 191. Slide 3 / 191. Algebraic Concepts. Table of Contents What is Algebra?
Slide 1 / 191 Slide 2 / 191 5th Grade Algebraic Concepts 2015-10-16 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 191 What is Algebra? Order of Operations Grouping
More informationUnit 5 Review Systems of Linear Equations and Inequalities
Unit 5 Review Systems of Linear Equations and Inequalities Name: Algebra 1B Day 1: Solutions to Systems and Solving by Graphing Warm Up: Determine if the point (2,5) is a solution to each of the systems
More informationWRITING EQUATIONS 4.1.1
WRITING EQUATIONS 4.1.1 In this lesson, students translate written information, often modeling everyday situations, into algebraic symbols and linear equations. Students use let statements to specifically
More informationName Period Date DRAFT
Name Period Date Equations and Inequalities Student Packet 4: Inequalities EQ4.1 EQ4.2 EQ4.3 Linear Inequalities in One Variable Add, subtract, multiply, and divide integers. Write expressions, equations,
More informationBuilding Concepts: Solving Systems of Equations Algebraically
Lesson Overview In this TI-Nspire lesson, students will investigate pathways for solving systems of linear equations algebraically. There are many effective solution pathways for a system of two linear
More informationExpressions & Equations Chapter Questions. 6. What are two different ways to solve equations with fractional distributive property?
Expressions & Equations Chapter Questions 1. Explain how distribution can simplify a problem. 2. What are like terms? 3. How do you combine like terms? 4. What are inverse operations? Name them. 5. How
More informationAlgebra. Chapter 6: Systems of Equations and Inequalities. Name: Teacher: Pd:
Algebra Chapter 6: Systems of Equations and Inequalities Name: Teacher: Pd: Table of Contents Chapter 6-1: SWBAT: Identify solutions of systems of linear equations in two variables; Solve systems of linear
More informationSystems of Equations Unit Five ONE NONE INFINITE
Systems of Equations Unit Five ONE NONE INFINITE Standards: 8.EE.8 Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables
More informationJune If you want, you may scan your assignment and convert it to a.pdf file and it to me.
Summer Assignment Pre-Calculus Honors June 2016 Dear Student: This assignment is a mandatory part of the Pre-Calculus Honors course. Students who do not complete the assignment will be placed in the regular
More informationAlgebra 1 Midterm Review
Name Block Algebra 1 Midterm Review MULTIPLE CHOICE Write the letter for the correct answer at the left of each question. 1. Solve: A. 8 C. 2. Solve: A. 43 C. 42 3. Solve the compound inequality and graph
More informationFinal Exam Study Guide
Algebra 2 Alei - Desert Academy 2011-12 Name: Date: Block: Final Exam Study Guide 1. Which of the properties of real numbers is illustrated below? a + b = b + a 2. Convert 6 yards to inches. 3. How long
More informationNOTES. [Type the document subtitle] Math 0310
NOTES [Type the document subtitle] Math 010 Cartesian Coordinate System We use a rectangular coordinate system to help us map out relations. The coordinate grid has a horizontal axis and a vertical axis.
More informationSOLVING LINEAR INEQUALITIES
Topic 15: Solving linear inequalities 65 SOLVING LINEAR INEQUALITIES Lesson 15.1 Inequalities on the number line 15.1 OPENER Consider the inequality x > 7. 1. List five numbers that make the inequality
More information1. Write an expression of the third degree that is written with a leading coefficient of five and a constant of ten., find C D.
1. Write an expression of the third degree that is written with a leading coefficient of five and a constant of ten. 2 2 2. If C = 4x 7x 9 and D = 5x 7x 3, find C D. 3. At an ice cream shop, the profit,,
More informationName Date. and y = 5.
Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five
More informationSections 8.1 & 8.2 Systems of Linear Equations in Two Variables
Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Department of Mathematics Porterville College September 7, 2014 Systems of Linear Equations in Two Variables Learning Objectives: Solve Systems
More information8th Grade. Slide 1 / 157. Slide 2 / 157. Slide 3 / 157. The Number System and Mathematical Operations Part 2. Table of Contents
Slide 1 / 157 Slide 2 / 157 8th Grade The Number System and Mathematical Operations Part 2 2015-11-20 www.njctl.org Table of Contents Slide 3 / 157 Squares of Numbers Greater than 20 Simplifying Perfect
More informationFSA Algebra I End-of-Course Review Packet. Algebra and Modeling
FSA Algebra I End-of-Course Review Packet Algebra and Modeling Table of Contents MAFS.912.A-APR.1.1 EOC Practice... 3 MAFS.912.A-CE1.1 EOC Practice... 5 MAFS.912.A-REI.2.3 EOC Practice... 7 MAFS.912.A-CE1.4
More informationRelationships Between Quantities
Algebra 1 Relationships Between Quantities Relationships Between Quantities Everyone loves math until there are letters (known as variables) in problems!! Do students complain about reading when they come
More informationChapter 1 Analytic Geometry
Chapter 1 Analytic Geometry 1. Find the coordinates of the midpoint of the line segment AB whose endpoints are A(5, 4) and B(-3, -4) 2. N (-2, 1) and D (8, 5) are the endpoints of the diameter of a circle.
More informationALGEBRA 2 Summer Review Assignments Graphing
ALGEBRA 2 Summer Review Assignments Graphing To be prepared for algebra two, and all subsequent math courses, you need to be able to accurately and efficiently find the slope of any line, be able to write
More informationChapter 7: Systems of Linear Equations
Chapter 7: Systems of Linear Equations Section 7.1 Chapter 7: Systems of Linear Equations Section 7.1: Developing Systems of Linear Equations Terminology: System of Linear Equations: A grouping of two
More informationALGEBRA 1. Unit 3 Chapter 6. This book belongs to: Teacher:
ALGEBRA 1 Teacher: Unit 3 Chapter 6 This book belongs to: UPDATED FALL 2016 1 2 Algebra 1 Section 6.1 Notes: Graphing Systems of Equations Day 1 Warm-Up 1. Graph y = 3x 1 on a coordinate plane. 2. Check
More informationAddition and Subtraction of real numbers (1.3 & 1.4)
Math 051 lecture notes Professor Jason Samuels Addition and Subtraction of real numbers (1.3 & 1.4) ex) 3 + 5 = ex) 42 + 29 = ex) 12-4 = ex) 7-9 = ex) -3-4 = ex) 6 - (-2) = ex) -5 - (-3) = ex) 7 + (-2)
More informationAlgebra 1 PAP Fall Exam Review
Name: Pd: 2016-2017 Algebra 1 PAP Fall Exam Review 1. A collection of nickels and quarters has a value of $7.30. The value of the quarters is $0.80 less than triple the value of the nickels. Which system
More informationBig Idea(s) Essential Question(s)
Middletown Public Schools Mathematics Unit Planning Organizer Subject Math Grade/Course Algebra I Unit 4 Linear Functions Duration 20 instructional days + 4 days reteaching/enrichment Big Idea(s) Essential
More informationUnit 7 Systems and Linear Programming
Unit 7 Systems and Linear Programming PREREQUISITE SKILLS: students should be able to solve linear equations students should be able to graph linear equations students should be able to create linear equations
More informationThis is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0).
This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/
More informationRate of Change and slope. Objective: To find rates of change from tables. To find slope.
Linear Functions Rate of Change and slope Objective: To find rates of change from tables. To find slope. Objectives I can find the rate of change using a table. I can find the slope of an equation using
More informationUNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES. Solving Equations and Inequalities in One Variable
UNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES This unit investigates linear equations and inequalities. Students create linear equations and inequalities and use them to solve problems. They
More information