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 Systems of Equations Solving Systems by Graphing Solving Systems by Substitution Solving Systems by Elimination Teacher Note Choosing your Strategy Writing Systems to Model Situations Solving Systems of Inequalities
Slide 4 / 179 8th Grade Review Return to Table of Contents Slide 5 / 179 When you have 2 or more linear equations that is called a system of equations, there will be two or more variables. To find the solution, you will need a set of two numbers (ordered pair) that makes all the equations true. You have previously learned how to solve a system using graphing, let's review. To solve by GRAPHING, you must graph both lines and find the point where they intersect. Slide 6 / 179 (3, 4) The solution to the system of equations will be the ordered pair: (3, 4)
Example: y = 2x + 3 y = -1x - 2 2 Slide 7 / 179 Step 1: Graph both lines from slope-intercept form on the same coordinate plane Step 2: Write the intersection point as an ordered pair. Given two sets of coordinate points that represent a system of linear equations, determine whether the lines intersect to given a solution to the system. Linear Equation 1: (1, 1) and (2, 3) Linear Equation 2: (1, -2) and (4, 4) Example Slide 8 / 179 Will the system of linear equations intersect into a solution? Example Decide if you will be able to find a solution to the system of equation just by inspecting. Do not try to solve algebraically. Slide 9 / 179 System: 6x + 3y = 10 6x + 3y = 5
Slide 10 / 179 Solving Systems by Graphing Return to Table of Contents Vocabulary A system of linear equations is two or more linear equations. Slide 11 / 179 The solution to a system of linear inequalities is the ordered pair that will satisfy both equations. One way to find the solution to a system is to graph the equations on the same coordinate plane and find the point of intersection. There are 3 different types of solutions that are possible to get when solving a system. They are easiest to understand by looking at the graph. Click here to watch a music video that introduces what we will learn about systems. Type 1: One Solution Slide 12 / 179 This is the most common type of solution, it happens when two lines intersect in exactly ONE place The slopes of the lines will be DIFFERENT
Compare the Slopes Slide 13 / 179 y= 2x + 5 6x + 2y = 4 m = 2-6x - 6x 2y = -6x + 4 2 2 2 y = -3x + 2 m = -3 What did we find out about the slopes? So, how many solutions will there be? Type 2: No Solution This happens when the lines NEVER intersect! The lines will be PARALLEL. Slide 14 / 179 The slopes of the lines will be THE SAME The y-intercepts will be DIFFERENT Compare the Slopes and Y-Intercepts Slide 15 / 179 y= -5x + 4 10x + 2y = 6 m = -5-10x - 10x b = 4 2y = -10x + 6 2 2 2 y = -5x + 3 m = -5 b = 3 What did we find out about the slopes and the y-intercepts? So, how many solutions will there be?
Type 3: Infinite Solutions This happens when the lines overlap! The lines will be the SAME EXACT line! Slide 16 / 179 The slopes of the lines will be THE SAME The y-intercepts will bethe SAME Compare the Slopes and Y-Intercepts Slide 17 / 179 y= 2x + 1-4x + 2y = 2 m = 2 + 4x + 4x b = 1 2y = 4x + 2 2 2 2 y = 2x + 1 m = 2 b = 1 What did we find out about the slopes and the y-intercepts? So, how many solutions will there be? How can you quickly decide the number of solutions a system has? Slide 18 / 179 1 Solution No Solution Infinitely Many Different slopes Different lines Same slope Different y-intercept Parallel Lines Same slope Same y-intercept Same Line
1 How many solutions does the following system have: y = 2x - 7 y = 3x + 8 Slide 19 / 179 A B C 1 solution no solution infinitely many solutions 2 How many solutions does the following system have: 3x - y = -2 y = 3x + 2 Slide 20 / 179 A B C 1 solution no solution infinitely many solutions 3 How many solutions does the following system have: 3x + 3y = 8 y = 1 3 x Slide 21 / 179 A B C 1 solution no solution infinitely many solutions
4 How many solutions does the following system have: y = 4x 2x - 0.5y = 0 Slide 22 / 179 A B C 1 solution no solution infinitely many solutions 5 How many solutions does the following system have: 3x + y = 5 6x + 2y = 1 Slide 23 / 179 A B C 1 solution no solution infinitely many solutions Consider this... Slide 24 / 179 Suppose you are walking to school. Your friend is blocks 5 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?
First, make a table to represent the problem. Solution Slide 25 / 179 Time (min.) Friend's distance from your start (blocks) Your distance from your start (blocks) 0 5 0 1 6 2 2 7 4 3 8 6 4 9 8 5 10 10 Next, plot the points on a graph. Solution Continued Slide 26 / 179 Time (min.) Friend's distance from your start (blocks) Your distance from your start(blocks) Blocks 0 5 0 1 6 2 2 7 4 3 8 6 4 9 8 5 10 10 Time (min.) Solution Continued Slide 27 / 179 The point where the lines intersect is the solution to the system. (5,10) is the solution Blocks In the context of the problem this means after 5 minutes, you will meet your friend at block 10. Time (min.)
Graphing Lines Slide 28 / 179 Recall from Algebra I that you need a minimum of two points to graph a line. Therefore, when solving a system of linear equations graphically, you will only need to plot two points for each equation. Solve the system of equations graphically: y = 2x -3 y = x - 1 Example Slide 29 / 179 Example Solve the following system by graphing: Slide 30 / 179 y = -3x + 4 y = x - 4
Checking Your Work Given the graph below, what is the point of intersection? y = -3x - 1 y = 4x + 6 (move the hand!) Slide 31 / 179 (-1, 2) Checking Your Work Slide 32 / 179 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) - 1 2 = 3-1 2 = 2 y = 4x + 6 2 = 4(-1) + 6 2 = -4 + 6 2 = 2 6 Solve the following system by graphing: Slide 33 / 179 y = -x + 4 y = 2x + 1 A (3, 1) B (1, 3) Click for choices AFTER students have graphed the system C (-1, 3) D (1, -3)
7 Solve the following system by graphing: Slide 34 / 179 A (0,-1) B (0,0) Click for choices AFTER students have graphed the system C (-1, 0) D (0, 1) 8 Solve the following system by graphing: Slide 35 / 179 y = x + 3 A (0, 4) B (-4, 2) C (5, 6) D (2, 5) Graphing Quickly Slide 36 / 179 Recall from 8th grade that slope-intercept form of a linear equation is: y = mx + b Where m = the slope and b = the y-intercept If you transform linear equations not in slope-intercept form to slope-intercept form, graphing them will be quicker.
Example Slide 37 / 179 Solve the following system of linear equations by graphing: 2x + y = 5 -x + y = 2 Step 1: Rewrite the linear equation in slope-intercept form 2x + y = 5-2x -2 x y = -2 x + 5 -x + y = 2 +x +x y = x + 2 Solution Continued Slide 38 / 179 Step 2: Plot the y-intercept and use the slope to plot the second point y-intercept = (0, 5) slope = -2 slope= (down 2, right 1) y-intercept = (0, 2) slope = 1 slope= (up 1, right 1) Solution Continued Step 3: Locate the Point of Intersection and check your work: (1, 3) Slide 39 / 179 y = -2 x + 5 3 = -2(1) + 5 3 = -2 + 5 y = x + 2 3 = 1 + 2 3 = 3 3 = 3
Example Slide 40 / 179 Solve the system of equations graphically: 2x + y = 3 x - 2y = 4 Step 1: Rewrite in slope-intercept form 2x + y = 3-2x -2 x y = -2 x + 3 x - 2y = 4 -x -x -2y = -x + 4-2 -2 1 y = x - 2 2 Solution Continued Step 2: Plot y-intercept and use slope to plot second point Slide 41 / 179 y-intercept = (0, 3) slope = -2 slope= (down 2, right 1) y-intercept = (0, -2) slope = slope= (up 1, right 2) Step 3: Locate the Point of Intersection and check your work: (2, -1) Solution Continued Step 3: Locate the Point of Intersection and check your work: (2, -1) Slide 42 / 179 y = -2 x + 3-1 = -2(2) + 3-1 = -4 + 3-1 = -1
9 What is the solution of the system of linear equations provided on the graph below? Slide 43 / 179 A (0, 1) B (1, 0) C (2, 3) D (3, 2) 10 Which graph below represents the solution to the following system of linear equations: Slide 44 / 179 A -x + 2y = 2 3y = x + 6 C B D 11 Solve the following system by graphing: Slide 45 / 179 A (3, 4) B (9, 2) Click for choices AFTER students have Cgraphed infintely the systemany D no solution
Example Slide 46 / 179 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 -18-3 -3 y = 3x + 6 Solution Continued Step 2: Plot y-intercept and use slope to plot second point Slide 47 / 179 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) Step 3: Locate the Point of Intersection and check your work: infinite amount of points: infinite solutions Example Slide 48 / 179 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 + 10-2 -2 y = 2x - 5 8x - 4y = 12-8x -8x -4y = -8x +12-4 -4 y = 2x -3
Solution Continued Step 2: Plot y-intercept and use slope to plot second point Slide 49 / 179 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) Step 3: Locate the Point of Intersection and check your work: no point of intersection: no solution 12 Solve the following system by graphing: Slide 50 / 179 y = 3x + 4 4y = 12x + 12 A (2, 4) B (0.4, 2.2) C D infinitely many no solution 13 Solve the following system by graphing: Slide 51 / 179 y = 3x + 4 4y = 12x + 16 A (3,4) B (-3,-4) C D infinitely many no solution
Slide 52 / 179 Solving Systems by Substitution Return to Table of Contents Example Solve the system of equations graphically. Slide 53 / 179 y = x + 6.1 y = -2x - 1.4 Note Why was it difficult Click to for solve Additional this Question system by graphing? Substitution Explanation Slide 54 / 179 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.
Solving by Substitution Slide 55 / 179 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) Example Slide 56 / 179 Solve the system using substitution: y = x + 6.1 y = -2x - 1.4 Step 1 : Choose an equation from the system and substitute it into the other equation y = x + 6.1 First Equation y = -2x - 1.4 Second Equation x + 6.1 = -2x - 1.4 Substitute First Equation into Second Equation Step 2: Solve the new equation Solution Continued Slide 57 / 179 x + 6.1 = -2x - 1.4 +2x -6.1 +2x - 6.1 3x = -7.5 x = -2.5 Step 3: Substitute the solution into either equation and solve y = x + 6.1 y = (-2.5) + 6.1 y = 3.6 The solution to the system of linear equations is (-2.5, 3.6)
Good Practice Slide 58 / 179 After you evaluate the solution, it is good practice is to check your work by substituting the solution into both equations. CHECK: See if (-2.5, 3.6) satisfies both equations y = -2x - 1.4 3.6 = -2(-2.5) - 1.4 3.6 = 5-1.4 3.6 = 3.6 y = x + 6.1 3.6 = -2.5 + 6.1 3.6 = 3.6 If your checks end in true statements, the solution is correct. Example Slide 59 / 179 Solve the system using substitution: 2x - 3y = -1 y = x - 1 Step 1: Substitute one equation into the other equation 2x - 3 y = -1 First Equation y = x - 1 Second Equation 2x - 3(x - 1) = -1 Substitution Solution Continued Slide 60 / 179 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 You end with the y = x - 1 2(4) - 3y = -1 correct with y = 4-1 8-3y = -1 either equation you y = 3-3y = -9 use for this step. y = 3 (4, 3) (4, 3)
Example Continued Slide 61 / 179 Check: See if (4, 3) satisfies both equations 2x - 3y = -1 y = x - 1 2(4) - 3(3) = -1 3 = 4-1 8-9 = -1 3 = 3-1 = -1 The ordered pair satisfies both equations so the solution is (4, 3) 14 Solve the system by substitution: y = x - 3 y = -x + 5 Slide 62 / 179 A (4, 9) B (-4, -9) Click for choices AFTER students have solved the system C (4, 1) D (1, 4) 15 Solve the system using substitution: Slide 63 / 179 A (2, -8) B (-3, 2) Click for choices AFTER students have solved the system C infinitely many solutions D no solutions
16 Solve the system using substitution. Slide 64 / 179 y = 4x - 11-4x + 3y = -1 A (4, 5) B (5, 4) Click for choices AFTER students have C infintely many solutions solved the system D no solutions Solve the system using substitution. Slide 65 / 179 17 y = 8x + 18 3x + 3y = 0 A (-2, -2) B (-2, 2) C (2, -2) Click for choices AFTER students have solved the system D (2, 2) Slide 66 / 179 Solve the system using substitution. 18 8x + 3y = -9 y = 3x + 14 A (-8, 5) B (7, 5) Click for choices AFTER students have solved the system C (-3, 5) D (-7, 5)
Choosing a Variable Slide 67 / 179 Examine each system of equations. Which variable would you choose to substitute? Why? y = 4x - 9.6 y = -2x + 9 -y + 4x = -1 x - 4y = 1 Note 2x + 4y = -10-8x - 3y = -12 19 Examine the system of equations below. Which variable could quickly be solved for and substituted into the other equation? Slide 68 / 179 y = -2x + 5 2y = 10-4x A B x y 20 Examine the system of equations below. Which variable could quickly be solved for and substituted into the other equation? Slide 69 / 179 2y - 8 = x y + 2x = 4 A B x y
21 Examine the system of equations below. Which variable could quickly be solved for and substituted into the other equation? Slide 70 / 179 x - y = 20 2x + 3y = 0 A B x y Rewriting Slide 71 / 179 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 Which letter is the easiest to solve for? The "y" in the first equation because there Click to discuss which letter. is only a "-1" as the coefficient. Solve for y: 3x - y = 5-3x -3x -y = -3x + 5-1 -1-1 y = 3x - 5 So, the original system is equivalent to: y = 3x - 5 click to see 2x + 5 y = -8 Solution Continued Slide 72 / 179 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
Solution Continued Slide 73 / 179 Substitute x = 1 into one of the equations. 2(1) + 5y = -8 2 + 5y = -8 5y = -10 y = -2 The ordered pair (1,-2) satisfies both equations in system. 3x - y = 5 2x + 5y = -8 3(1) - (-2) = 5 2(1) + 5(-2) = -8 3 + 2 = 5 2-10 = -8 5 = 5-8 = -8 Solve using substitution. Slide 74 / 179 22 6x + y = 6-3x + 2y = -18 A (-6, 2) B (6, -2) Click for choices AFTER students have C (-6, -2) solved the system D (2, -6) Solve using substitution. Slide 75 / 179 23 2x - 8y = 20 -x + 6y = -12 A (6, -1) B (-6, 5) Click for choices AFTER students have C (5, 5) solved the system D (-6, -1)
Solve using substitution. Slide 76 / 179 24-3x - 3y = 12-4x - 7y = 7 A (-3, -7) B (-7, 3) Click for choices AFTER students have C (3, 7) solved the system D (7, 3) Example Slide 77 / 179 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 for the trip? Let v = the number of vans and c = the number of cars Set up the system: Slide 78 / 179 Drivers: v + c = 4 People: 6v + 4c = 22 Solve the system by substitution: v + c = 4 -solve the first equation for v v = -c + 4 -substitute -c + 4 for v in the 6(-c + 4) + 4c = 22 second equation -6c + 24 + 4 c = 22 -solve for c -2c + 24 = 22-2c = -2 c = 1 v + c = 4 v + 1 = 4 v = 3 -substitute c = 1 in the 1st equation -solve for v
Solution Slide 79 / 179 Since c = 1 and v = 3, they should use 1 car and 3 vans. Check the solution in both equations: v + c = 4 6v + 4c = 22 3 + 1 = 4 6(3) + 4(1) = 22 4 = 4 18 + 4 = 22 Example Slide 80 / 179 Solve this system using substitution: x + y = 6 5x + 5y = 10 x + y = 6 -solve the first equation for x x = 6 - y 5(6 - y) + 5y = 10 -substitute 6 - y for x in 2nd equation 30-5y + 5y = 10 -solve for y 30 = 10 -This is FALSE! Since 30 = 10 is a false statement, the system has no solution. Answer: NO SOLUTION Solve the following system using substitution: x + 4y = -3 2x + 8y = -6 Example Slide 81 / 179 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
25 Solve the system by substitution: y = x - 6 y = -4 Slide 82 / 179 A (-10, -4) B (-4, 2) C (2, -4) D (10, 4) 26 Solve the system by substitution: y + 2x = -14 y = 2x + 18 Slide 83 / 179 A (1, 20) B (1, 18) C (8, -2) D (-8, 2) 27 Solve the system by substitution: 4x = -5y + 50 x = 2y - 7 Slide 84 / 179 A (6, 6.5) B (5, 6) C (4, 5) D (6, 5)
28 Solve the system by substitution: y = -3x + 23 -y + 4x = 19 Slide 85 / 179 A (6, 5) B (-7, 5) Click for choices AFTER students have solved the system C (42, -103) D (6, -5) 29 Solve the system using substitution. Slide 86 / 179 A (-4, 5) B (4, -1) Click for choices AFTER students have solved the system C infinitely many solutions D no solutions 30 Solve using substitution. Slide 87 / 179 16x + 2y = -5 y = -8x - 6 A (-3, -1) B C No Click Solution for choices AFTER students have solved the system Infinite Solutions D (-1, -3)
Slide 88 / 179 Solving System by Elimination Return to Table of Contents Standard Form Slide 89 / 179 Recall that the Standard Form of a linear equation is: Ax + By = C When both linear equations of a system are in s tandard form the system can be solved by using elimination. The elimination strategy adds or subtracts the equations in the system to eliminate a variable. Additive Inverses Slide 90 / 179 Let's talk about what's happening with these numbers - 2 + 2 = 3 + (-3)= -5x + 5x = 9x + (-9x) =
Choosing a Variable Slide 91 / 179 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. Addition or Subtraction Slide 92 / 179 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. { 3x { Opposite Coefficients -3x Add 3x + (-3x) 0x Solve the following system by elimination: Example Slide 93 / 179 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 = 10
Solution Continued Step 3: Substitute the solution into either equation and solve Slide 94 / 179 x = 10 5(10) + y = 44 50 + y = 44 y = -6 The solution to the system is (10, -6) Check: 5x + y = 44 5(10) + (-6) = 44 50-6 = 44 44 = 44-4x - y = -34-4(10) - (-6) = -34-40 + 6 = -34-34 = -34 Solve the following system by elimination: 3x + y = 15-3x - 3y = -21 Example Slide 95 / 179 Step 1: Choose which variable to eliminate 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 Solution Continued Step 3: Substitute the solution into either equation and solve Slide 96 / 179 y = 3 3x + 3 = 15 3x = 12 x = 4 The solution to the system is (4, 3) Check: 3x + y = 15 3(4) + 3 = 15 12 + 3 = 15 15 = 15-3x - 3y = -21-3(4) - 3(3) = -21-12 - 9 = -21-21 = -21
31 Solve the system by elimination: Slide 97 / 179 A (5, 1) B (-5, -1) Click for choices AFTER students have solved the system C (1, 5) x + y = 6 x - y = 4 D no solution 32 Solve the system by elimination: Slide 98 / 179 2x + y = -5 2x - y = -3 A (-2,1) B (-1,-2) C (-2,-1) D infinitely many 33 Solve using elimination. -2x - 8y = 10 2x - 6y = 18 Slide 99 / 179 A (-2, 3) B (4, -6) Click for choices AFTER students have C solved (-6, 4) the system D (3, -2)
Multiple Methods Slide 100 / 179 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 Solution Continued Slide 101 / 179 First Method -1(-2x + y = -4) = 2x - y = 4 5x + y = 17 2x - y = 4 7x = 21 Second Method 5x + y = 17 -(-2x + y = -4) 7x = 21 x = 3 x = 3 Both methods produce the same solution because multiplying by -1 then adding is the same as subtracting the entire equation. Solution Continued Step 3: Substitute the solution into either equation and solve Slide 102 / 179 x = 3-2(3) + y = -4-6 + y = -4 y = 2 The solution to the system is (3, 2) Check: 5x + y = 17 5(3) + 2 = 17 15 + 2 = 17 17 = 17-2x + y = -4-2(3) + 2 = -4-6 + 2 = -4-4 = -4
34 Solve the system by elimination: 2x + y = -6 3x + y = -10 Slide 103 / 179 A (-4, 2) B (3, 5) C (4, 2) D infinitely many 35 Solve the system by elimination: 3x + 6y = 48-5x + 6y = 32 Slide 104 / 179 A (2, -7) B (2, 7) C (7, 2) D infinitely many Common Coefficient Slide 105 / 179 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.
Solve the following system using elimination: Example Slide 106 / 179 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 = -10 + 10x - 4y = 36 13x = 26 x = 2 Example Continued Solve for y, by substituting x = 2 into one of the equations. 3x + 4y = -10 3(2) + 4y = -10 6 + 4y = -10 4y = -16 y = -4 (2,-4) is the solution Slide 107 / 179 Check: 3x + 4y = -10 3(2) + 4(-4) = -10 6 + -16 = -10-10 = -10 5x - 2y = 18 5(2) - 2(-4) = 18 10 + 8 = 18 18 = 18 Choosing Variable to Eliminate Slide 108 / 179 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.
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 Now solve by subtracting the equations. 15x + 20y = -50 -(15x - 6y = 54) 26y = -104 y = -4 Example Slide 109 / 179 3(5x - 2y = 18) 15x - 6y = 54 Example Continued Solve for x, by substituting y = -4 into one of the equations. 3x + 4y = -10 3x + 4(-4) = -10 3x + -16 = -10 3x = 6 x = 2 (2,-4) is the solution. Slide 110 / 179 Check: 3x + 4y = -10 3(2) + 4(-4) = -10 6 + -16 = -10-10 = -10 5x - 2y = 18 5(2) - 2(-4) = 18 10 + 8 = 18 18 = 18 Examine each system of equations. Which variable would you choose to eliminate? What do you need to multiply each equation by? Slide 111 / 179 2x + 5y = -1 x + 2y = 0 3x + 8y = 81 5x - 6y = -39 Note 3x + 6y = 6 2x - 3y = 4
36 Which variable can you eliminate with the least amount of work in the system below? Slide 112 / 179 2x + 5y = 20 3x - 10y = 37 A x B y 37 Solve the following system of equations using elimination: Slide 113 / 179 2x + 5y = 20 3x - 10y = 37 A (1, 57) B (1, 77) C D infinitely many solutions 38 Which variable can you eliminate with the least amount of work in the system below? Slide 114 / 179 x + 3y = 4 3x + 4y = 2 A x B y
39 What will you multiply the first equation by in order to solve this system using elimination? Slide 115 / 179 x + 3y = 4 3x + 4y = 2 Slide 116 / 179 Example Slide 117 / 179 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 = 8 + -18x + 10y = 10 0 = 18 is this true? False, NO SOLUTION Move for solution
Example Slide 118 / 179 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 = -22 + 4x +10y = 22 0 = 0 is this true? True, INFINITE SOLUTIONS Move for solution 41 Solve the system by elimination: x - y = 5 x - y = -7 Slide 119 / 179 A (11, -4) B (4, 11) Click for choices AFTER students have solved the system C (-4, -11) D no solution 42 Solve using elimination. -20x - 18y = -28 10x + 9y = 14 Slide 120 / 179 A (-8, -1) B C infinite Click for choices solutions AFTER students have no solution solved the system D (-1, 8)
43 Solve using elimination. 9x + 3y = 27 18 + 6y = 30 Slide 121 / 179 A infinite solutions B (4, 7) Click for choices AFTER students have C (-7, solved 4) the system D no solution Slide 122 / 179 Choose Your Strategy Return to Table of Contents Choosing Strategy Slide 123 / 179 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
Example Slide 124 / 179 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? Step 1: Define your variables Example Continued Slide 125 / 179 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 = 470 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 Note common coefficient. Example Continued Slide 126 / 179 a = 89 a + s = 292 89 + s = 292 s = 203 There were 89 adult tickets and 203 student tickets sold Check: a + s = 292 89 + 203 = 292 292 = 292 3a + s = 470 3(89) + 203 = 470 267 + 203 = 470 470 = 470
44 What method would require the least amount of work to solve the following system: Slide 127 / 179 y = 3x - 1 y = 4x A B C graphing substitution elimination 45 Solve the following system of linear equations using the method of your choice: Slide 128 / 179 A (-4, -1) B (-1, -4) C (-1, 4) D (1, 4) y = 3x - 1 y = 4x 46 What method would require the least amount of work to solve the following system: Slide 129 / 179 4s - 3t = 8 t = -2s -1 A B C graphing substitution elimination
Slide 130 / 179 48 What method would require the least amount of work to solve the following system: Slide 131 / 179 3m - 4n = 1 3m - 2n = -1 A B C graphing substitution elimination 49 Solve the following system of linear equations using the method of your choice: Slide 132 / 179 3m - 4n = 1 3m - 2n = -1 A (-2, -1) B (-1, -1) C (-1, 1) D (1, 1)
50 What method would require the least amount of work to solve the following system: Slide 133 / 179 A B C graphing substitution elimination 51 Solve the following system of linear equations using the method of your choice: Slide 134 / 179 y = -x A (-6, 12) B (2, -4) Click for choices AFTER students have solved the system C (-2, 2) D (1, -2) 52 What method would require the least amount of work to solve the following system: Slide 135 / 179 u = 4v 3u - 3v = 7 A B C graphing substitution elimination
53 Solve the following system of linear equations using the method of your choice: Slide 136 / 179 u = 4v 3u - 3v = 7 A C (28, 7) B D 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 Slide 137 / 179 A B C t = 99-3.5m t = 0 + 2.5m t = 99 + 3.5m t = 0 + 2.5m t = 99 + 3.5m t = 0-2.5m 55 Which method would you use to solve the system from the previous question? Slide 138 / 179 t = 99-3.5m t = 0 + 2.5m A B C graphing substitution elimination
56 Solve the following system of linear equations: Slide 139 / 179 t = 99-3.5m Click to Reveal System t = 0 + 2.5m A m = 1 t = 2.5 B m = 1 t = 95.5 C m = 16.5 t = 6.6 D m = 16.5 t = 41.25 57 Choose a strategy and then 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? Slide 140 / 179 A 1 B -1 C 3 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. Slide 141 / 179 Writing Systems to Model Situations Return to Table of Contents
Creating and Solving Systems Slide 142 / 179 Step 1: Define the variables Step 2: Analyze components and create equations Step 3: Solve the system utilizing the best strategy Example Slide 143 / 179 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 = 148 12a + 9c = 1,410 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. Example Continued Slide 144 / 179 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 = 148 12a + 9c = 1,410 (1) c = -a + 148 (2) 12a + 9(-a + 148) = 1410 12a - 9a + 1332 = 1410 a + c = 148 26 + c = 148 c = 122 3a = 78 a = 26
Example Slide 145 / 179 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 = 6.00 2p + 3c = 5.25 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. Example Continued Slide 146 / 179 3p + 2c = 6.00 2p + 3c = 5.25 Elimination: Multiply first equation by 2 Multiply second equation by -3 6p + 4c = 12-6p - 9c = -15.75-5c = -3.75 c = 0.75 Cola: $0.75 Pizza: $1.50 3p + 2c = 6.00 3p + 2(0.75) = 6 3p + 1.5 = 6 3p = 4.5 p = 1.5 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. Slide 147 / 179 How many packages of cards were sold? You will how many packages of gift wrap in the next question.
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. Slide 148 / 179 How many packages of gift wrap were sold? 60 The sum of two numbers is 47, and their difference is 15. What is the larger number? Slide 149 / 179 A 16 B 31 C 32 D 36 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 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? Slide 150 / 179 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011.
62 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many quarters do you have? Slide 151 / 179 63 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many nickels do you have? Slide 152 / 179 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? Slide 153 / 179 A $0.50 B $0.75 C $1.00 D $2.00 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011.
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? Slide 154 / 179 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 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? Slide 155 / 179 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. Slide 156 / 179 Solving Systems of Inequalities Return to Table of Contents
Vocabulary Slide 157 / 179 A system of linear inequalities is two or more linear inequalities. The solution to a system of linear inequalities is the intersection of the half-planes formed by each linear inequality. The most direct way to find the solution to a system of linear inequalities is to graph the equations on the same coordinate plane and find the region of intersection. Graphing a System of Linear Inequalities Slide 158 / 179 Step 1: Graph the boundary lines of each inequality. Remember: dashed line for < and > solid line for < and > Step 2: Shade the half-plane for each inequality. Step 3: Identify the intersection of the half-planes. This is the solution to the system of linear inequalities. Example Solve the following system of linear inequalities. Slide 159 / 179 Step 1: y < -1x + 3 2 y < 1x 4
Step 2: Example Continued y < -1x + 3 2 y < 1x 4 Slide 160 / 179 Step 3: Example Continued y < -1x + 3 2 y < 1x 4 Slide 161 / 179 Solve the following system of linear inequalities. Step 1: Example Slide 162 / 179 2x + y > -4 x - 2y < 4
Example Continued Slide 163 / 179 2x + y > -4 x - 2y < 4 Step 2: Example Continued 2x + y > -4 x - 2y < 4 Slide 164 / 179 Step 3: Example Solve the following system of linear inequalities. Slide 165 / 179 4x + 2y < 8 4x + 2y > -8 Step 1:
Example Continued Slide 166 / 179 4x + 2y < 8 4x + 2y > -8 Step 2: Example Continued Slide 167 / 179 4x + 2y < 8 4x + 2y > -8 Step 3: Solve the following system of linear inequalities. Example Slide 168 / 179 y < 3 Step 1: x > 1
Example Continued Slide 169 / 179 y < 3 Step 2: x > 1 Example Continued Slide 170 / 179 y < 3 Step 3: x > 1 67 Choose the graph below that displays the solution to the following system of linear inequalities: Slide 171 / 179 y > -2x + 1 y < x + 2 A B C
68 Choose the graph below that displays the solution to the following system of linear inequalities: Slide 172 / 179 x > 2 y < 5 A B C 69 Choose the graph below that displays the solution to the following system of linear inequalities: Slide 173 / 179-5x + y > -2 4x + y < 1 A B C 70 Choose the graph below that displays the solution to the following system of linear inequalities: Slide 174 / 179 3x + 2y < 12 2x - 2y < 20 A B C
71 Choose all of the linear inequalities that correspond to the following graph: Slide 175 / 179 A y > -2 B y < 2 C 3x + 4y > 12 D 3x + 4y < 12 72 Which point is in the solution set of the system of Slide 176 / 179 inequalities shown in the accompanying graph? A (0, 4) C (-4, 1) B (2, 4) D (4, -1) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 73 Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph? Slide 177 / 179 A (0, 0) B (0, 1) C (1, 5) D (3, 2) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011.
74 Which ordered pair is in the solution set of the following system of linear inequalities? Slide 178 / 179 y < 2x + 2 y x 1 A (0, 3) B (2, 0) C ( 1, 0) D ( 1, 4) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. 75 Mr. Braun has $75.00 to spend on pizzas and soda for a picnic. Pizzas cost $9.00 each and the drinks cost $0.75 each. Five times as many drinks as pizzas are needed. What is the maximum number of pizzas that Mr. Braun can buy? Slide 179 / 179 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011.