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 of Equations Solving Systems by Graphing Slide 4 / 179 8th Grade Review Solving Systems by Substitution Solving Systems by Elimination Teacher Note Choosing your Strategy Writing Systems to Model Situations Solving Systems of Inequalities 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. Slide 6 / 179 To solve by GRPHING, you must graph both lines and find the point where they intersect. (3, 4) The solution to the system of equations will be the ordered pair: (3, 4) You have previously learned how to solve a system using graphing, let's review.
: y = 2x + 3 Step 1: y = -1x - 2 2 Graph both lines from slope-intercept form on the same coordinate plane Slide 7 / 179 Slide 8 / 179 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) Step 2: Write the intersection point as an ordered pair. Will the system of linear equations intersect into a solution? Slide 9 / 179 Decide if you will be able to find a solution to the system of equation just by inspecting. Do not try to solve algebraically. System: 6x + 3y = 10 6x + 3y = 5 Slide 10 / 179 Solving Systems by Graphing Return to Table of Contents Slide 11 / 179 Vocabulary system of linear equations is two or more linear equations. 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 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. The slopes of the lines will be DIFFERENT 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.
Slide 13 / 179 Compare the Slopes 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? Type 2: No Solution Slide 14 / 179 This happens when the lines NEVER intersect! The lines will be PRLLEL. The slopes of the lines will be THE SME The y-intercepts will be DIFFERENT So, how many solutions will there be? Slide 15 / 179 Compare the Slopes and Y-Intercepts 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? Slide 16 / 179 Type 3: Infinite Solutions This happens when the lines overlap! The lines will be the SME EXCT line! The slopes of the lines will be THE SME The y-intercepts will bethe SME So, how many solutions will there be? Slide 17 / 179 Compare the Slopes and Y-Intercepts 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? Slide 18 / 179 How can you quickly decide the number of solutions a system has? 1 Solution No Solution Infinitely Many Different slopes Different lines Same slope Different y-intercept Parallel Lines Same slope Same y-intercept Same Line
Slide 19 / 179 1 How many solutions does the following system have: 1 solution y = 2x - 7 y = 3x + 8 Slide 20 / 179 2 How many solutions does the following system have: 3x - y = -2 y = 3x + 2 C no solution infinitely many solutions 1 solution no solution C infinitely many solutions Slide 21 / 179 Slide 22 / 179 3 How many solutions does the following system have: 3x + 3y = 8 y = 1 3 x 4 How many solutions does the following system have: y = 4x 2x - 0.5y = 0 1 solution 1 solution C no solution infinitely many solutions C no solution infinitely many solutions Slide 23 / 179 Slide 24 / 179 5 How many solutions does the following system have: 3x + y = 5 6x + 2y = 1 Consider this... 1 solution no solution 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? C infinitely many solutions
Slide 25 / 179 Solution First, make a table to represent the problem. Next, plot the points on a graph. Slide 26 / 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 locks 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 Slide 27 / 179 The point where the lines intersect is the solution to the system. Time (min.) Slide 28 / 179 Graphing Lines Recall from lgebra I that you need a minimum of two points to graph a line. (5,10) is the solution locks In the context of the problem this means after 5 minutes, you will meet your friend at block 10. Therefore, when solving a system of linear equations graphically, you will only need to plot two points for each equation. Time (min.) Slide 29 / 179 Solve the system of equations graphically: y = 2x -3 y = x - 1 Slide 30 / 179 Solve the following system by graphing: y = -3x + 4 y = x - 4
Slide 31 / 179 Checking Your Work Given the graph below, what is the point of intersection? y = -3x - 1 y = 4x + 6 (move the hand!) Slide 32 / 179 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 OTH lines. (-1, 2) (-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 Slide 33 / 179 6 Solve the following system by graphing: Slide 34 / 179 7 Solve the following system by graphing: y = -x + 4 y = 2x + 1 (3, 1) (1, 3) Click for choices graphed the system C (-1, 3) D (1, -3) (0,-1) (0,0) Click for choices graphed the system C (-1, 0) D (0, 1) Slide 35 / 179 8 Solve the following system by graphing: y = x + 3 (0, 4) (-4, 2) C (5, 6) D (2, 5) Slide 36 / 179 Graphing Quickly 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.
Slide 37 / 179 Solve the following system of linear equations by graphing: 2x + y = 5 -x + y = 2 y-intercept = (0, 5) slope = -2 slope= (down 2, right 1) Slide 38 / 179 Step 2: Plot the y-intercept and use the slope to plot the second point 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 y-intercept = (0, 2) slope = 1 slope= (up 1, right 1) Slide 39 / 179 Step 3: Locate the Point of Intersection and check your work: (1, 3) y = -2 x + 5 y = x + 2 3 = -2(1) + 5 3 = 1 + 2 3 = -2 + 5 3 = 3 3 = 3 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 Slide 41 / 179 Step 2: Plot y-intercept and use slope to plot second point y-intercept = (0, 3) slope = -2 slope= (down 2, right 1) y-intercept = (0, -2) slope = slope= (up 1, right 2) Slide 42 / 179 Step 3: Locate the Point of Intersection and check your work: (2, -1) y = -2 x + 3-1 = -2(2) + 3-1 = -4 + 3-1 = -1 Step 3: Locate the Point of Intersection and check your work: (2, -1)
Slide 43 / 179 9 What is the solution of the system of linear equations provided on the graph below? Slide 44 / 179 10 Which graph below represents the solution to the following system of linear equations: -x + 2y = 2 3y = x + 6 C (0, 1) C (2, 3) D (1, 0) D (3, 2) Slide 45 / 179 11 Solve the following system by graphing: Slide 46 / 179 Solve the system of equations graphically: y = 3x + 6 9x - 3y = -18 (3, 4) (9, 2) Click for choices Cgraphed infintely the systemany D no solution Step 1: Rewrite in slope-intercept form y = 3x + 6 9x - 3y = -18-9x -9x -3y = -9x -18-3 -3 y = 3x + 6 Slide 47 / 179 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) Slide 48 / 179 Solve the system of equations graphically: 4x - 2y = 10 8x - 4y = 12 Step 1: Rewrite in slope-intercept form 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 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
Slide 49 / 179 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) Step 3: Locate the Point of Intersection and check your work: no point of intersection: no solution Slide 51 / 179 Slide 50 / 179 12 Solve the following system by graphing: y = 3x + 4 4y = 12x + 12 (2, 4) (0.4, 2.2) C infinitely many D no solution Slide 52 / 179 13 Solve the following system by graphing: y = 3x + 4 4y = 12x + 16 Solving Systems by Substitution (3,4) (-3,-4) C infinitely many D no solution Return to Table of Contents Slide 53 / 179 Solve the system of equations graphically. y = x + 6.1 y = -2x - 1.4 Slide 54 / 179 Substitution Explanation Graphing can be inefficient or approximate. Note nother way to solve a system of linear equations is to use substitution. Substitution allows you to create a one variable equation. Why was it difficult Click to for solve dditional this Question system by graphing?
Slide 55 / 179 Solving by Substitution Step 1: If you are not given a variable already alone, find the ESIEST variable to solve for (get it alone) Slide 56 / 179 Solve the system using substitution: y = x + 6.1 y = -2x - 1.4 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) 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 Slide 57 / 179 Step 2: Solve the new equation x + 6.1 = -2x - 1.4 +2x -6.1 +2x - 6.1 3x = -7.5 x = -2.5 Slide 58 / 179 Good Practice fter 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 Step 3: Substitute the solution into either equation and solve y = x + 6.1 y = (-2.5) + 6.1 y = 3.6 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 The solution to the system of linear equations is (-2.5, 3.6) If your checks end in true statements, the solution is correct. Slide 59 / 179 Solve the system using substitution: 2x - 3y = -1 y = x - 1 Slide 60 / 179 Step 2: Solve the new equation 2x - 3(x - 1) = -1 2x - 3x + 3 = -1 x = 4 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 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)
Slide 61 / 179 Slide 62 / 179 Continued Check: See if (4, 3) satisfies both equations 14 Solve the system by substitution: y = x - 3 y = -x + 5 2x - 3y = -1 y = x - 1 2(4) - 3(3) = -1 8-9 = -1-1 = -1 3 = 4-1 3 = 3 (4, 9) (-4, -9) Click for choices solved the system C (4, 1) The ordered pair satisfies both equations so the solution is (4, 3) D (1, 4) Slide 63 / 179 Slide 64 / 179 15 Solve the system using substitution: 16 Solve the system using substitution. y = 4x - 11-4x + 3y = -1 (2, -8) (-3, 2) Click for choices solved the system C infinitely many solutions D no solutions (4, 5) (5, 4) Click for choices C infintely many solutions solved the system D no solutions Slide 65 / 179 Slide 66 / 179 Solve the system using substitution. 17 y = 8x + 18 3x + 3y = 0 Solve the system using substitution. 18 8x + 3y = -9 y = 3x + 14 (-2, -2) (-2, 2) C (2, -2) Click for choices solved the system (-8, 5) (7, 5) Click for choices solved the system C (-3, 5) D (2, 2) D (-7, 5)
Slide 67 / 179 Examine each system of equations. Which variable would you choose to substitute? Why? y = 4x - 9.6 y = -2x + 9 Choosing a Variable Slide 68 / 179 19 Examine the system of equations below. Which variable could quickly be solved for and substituted into the other equation? y = -2x + 5 2y = 10-4x -y + 4x = -1 x - 4y = 1 Note x y 2x + 4y = -10-8x - 3y = -12 Slide 69 / 179 20 Examine the system of equations below. Which variable could quickly be solved for and substituted into the other equation? 2y - 8 = x y + 2x = 4 Slide 70 / 179 21 Examine the system of equations below. Which variable could quickly be solved for and substituted into the other equation? x - y = 20 2x + 3y = 0 x y x y Slide 71 / 179 Rewriting Sometimes you need to rewrite one of the equations so that you can use the substitution method. For example: The system: Which letter is the easiest to solve for? 3x - y = 5 The "y" in the first equation because there Click to discuss which letter. 2x + 5y = -8 is only a "-1" as the coefficient. Solve for y: So, the original system is equivalent to: 3x - y = 5 y = 3x - 5 click to see -3x -3x 2x + 5 y = -8 -y = -3x + 5-1 -1-1 y = 3x - 5 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
Slide 73 / 179 Slide 74 / 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. 22 6x + y = 6-3x + 2y = -18 (-6, 2) (6, -2) Click for choices C (-6, -2) solved the system D (2, -6) Solve using substitution. 23 2x - 8y = 20 -x + 6y = -12 (6, -1) (-6, 5) Click for choices C (5, 5) solved the system D (-6, -1) Slide 75 / 179 Solve using substitution. 24-3x - 3y = 12-4x - 7y = 7 (-3, -7) (-7, 3) Click for choices C (3, 7) solved the system D (7, 3) Slide 76 / 179 Slide 77 / 179 Set up the system: Drivers: v + c = 4 People: 6v + 4c = 22 Slide 78 / 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 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
Slide 79 / 179 Slide 80 / 179 Solution Since c = 1 and v = 3, they should use 1 car and 3 vans. Solve this system using substitution: x + y = 6 5x + 5y = 10 Check the solution in both equations: v + c = 4 6v + 4c = 22 3 + 1 = 4 6(3) + 4(1) = 22 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 FLSE! 4 = 4 18 + 4 = 22 Since 30 = 10 is a false statement, the system has no solution. nswer: NO SOLUTION Slide 81 / 179 Solve the following system using substitution: x + 4y = -3 2x + 8y = -6 Slide 82 / 179 25 Solve the system by substitution: y = x - 6 y = -4 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 LWYS TRUE! (-10, -4) (-4, 2) Since -6 = -6 is always a true statement, there are infinitely many solutions to the system. nswer: Infinite Solutions C (2, -4) D (10, 4) Slide 83 / 179 Slide 84 / 179 26 Solve the system by substitution: y + 2x = -14 y = 2x + 18 27 Solve the system by substitution: 4x = -5y + 50 x = 2y - 7 (1, 20) (6, 6.5) (1, 18) C (8, -2) (5, 6) C (4, 5) D (-8, 2) D (6, 5)
Slide 85 / 179 28 Solve the system by substitution: y = -3x + 23 -y + 4x = 19 Slide 86 / 179 29 Solve the system using substitution. (6, 5) (-7, 5) Click for choices solved the system C (42, -103) D (6, -5) (-4, 5) (4, -1) Click for choices solved the system C infinitely many solutions D no solutions Slide 87 / 179 Slide 88 / 179 30 Solve using substitution. 16x + 2y = -5 y = -8x - 6 (-3, -1) Solving System by Elimination C No Click Solution for choices solved the system Infinite Solutions D (-1, -3) Return to Table of Contents Slide 89 / 179 Standard Form Recall that the Standard Form of a linear equation is: x + y = C Slide 90 / 179 dditive Inverses Let's talk about what's happening with these numbers - 2 + 2 = 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. 3 + (-3)= -5x + 5x = 9x + (-9x) =
Slide 91 / 179 Choosing a Variable Slide 92 / 179 ddition or Subtraction 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. If the variables have the same coefficient, subtract the two equations to eliminate the variable. { Same Coefficients 3x 3x Subtract { 3x -(3x) If the variables have opposite coefficients, add the two equations to eliminate the variable. 0x { 3x { Opposite Coefficients -3x dd 3x + (-3x) 0x Slide 93 / 179 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 Slide 94 / 179 Step 3: Substitute the solution into either equation and solve x = 10 5(10) + y = 44 50 + y = 44 y = -6 The solution to the system is (10, -6) Step 2: dd the two equations 5x + y = 44-4x - y = -34 x + 0y = 10 x = 10 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 Slide 95 / 179 Solve the following system by elimination: 3x + y = 15-3x - 3y = -21 Step 1: Choose which variable to eliminate The x in both equations have opposite coefficients so they will be the easiest to eliminate Slide 96 / 179 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) Step 2: dd the two equations 3x + y = 15-3x - 3y = -21-2y = -6 y = 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
Slide 97 / 179 31 Solve the system by elimination: Slide 98 / 179 32 Solve the system by elimination: x + y = 6 x - y = 4 2x + y = -5 2x - y = -3 (5, 1) (-5, -1) Click for choices solved the system C (1, 5) (-2,1) (-1,-2) C (-2,-1) D no solution D infinitely many 33 Solve using elimination. -2x - 8y = 10 2x - 6y = 18 Slide 99 / 179 Slide 100 / 179 Multiple Methods There are 2 ways to complete the problem below using elimination. 5x + y = 17-2x + y = -4 (-2, 3) (4, -6) Click for choices C solved (-6, 4) the system D (3, -2) 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: dd 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 Slide 101 / 179 First Method Second Method -1(-2x + y = -4) = 2x - y = 4 5x + y = 17 -(-2x + y = -4) 5x + y = 17 7x = 21 2x - y = 4 7x = 21 x = 3 Slide 102 / 179 Step 3: Substitute the solution into either equation and solve x = 3-2(3) + y = -4-6 + y = -4 y = 2 The solution to the system is (3, 2) x = 3 oth methods produce the same solution because multiplying by -1 then adding is the same as subtracting the entire equation. 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
Slide 103 / 179 Slide 104 / 179 34 Solve the system by elimination: 35 Solve the system by elimination: 2x + y = -6 3x + 6y = 48 3x + y = -10-5x + 6y = 32 (-4, 2) (3, 5) (2, -7) (2, 7) C (4, 2) C (7, 2) D infinitely many D infinitely many Slide 105 / 179 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. Slide 106 / 179 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 = -10 + 10x - 4y = 36 13x = 26 x = 2 Slide 107 / 179 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 Check: 3x + 4y = -10 5x - 2y = 18 3(2) + 4(-4) = -10 5(2) - 2(-4) = 18 6 + -16 = -10 10 + 8 = 18-10 = -10 18 = 18 Slide 108 / 179 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.
Slide 109 / 179 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 Slide 110 / 179 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. Now solve by subtracting the equations. 15x + 20y = -50 -(15x - 6y = 54) 26y = -104 y = -4 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 Slide 111 / 179 Slide 112 / 179 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 36 Which variable can you eliminate with the least amount of work in the system below? 2x + 5y = 20 3x - 10y = 37 3x + 8y = 81 5x - 6y = -39 Note x y 3x + 6y = 6 2x - 3y = 4 Slide 113 / 179 Slide 114 / 179 37 Solve the following system of equations using elimination: 38 Which variable can you eliminate with the least amount of work in the system below? (1, 57) 2x + 5y = 20 3x - 10y = 37 x + 3y = 4 3x + 4y = 2 (1, 77) C x y D infinitely many solutions
Slide 115 / 179 Slide 116 / 179 39 What will you multiply the first equation by in order to solve this system using elimination? x + 3y = 4 3x + 4y = 2 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 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 Slide 119 / 179 41 Solve the system by elimination: x - y = 5 x - y = -7 42 Solve using elimination. -20x - 18y = -28 10x + 9y = 14 Slide 120 / 179 (-8, -1) (11, -4) (4, 11) Click for choices solved the system C (-4, -11) infinite Click for choices solutions C no solution solved the system D (-1, 8) D no solution
Slide 121 / 179 Slide 122 / 179 43 Solve using elimination. 9x + 3y = 27 18 + 6y = 30 infinite solutions (4, 7) Click for choices C (-7, solved 4) the system Choose Your Strategy D no solution Return to Table of Contents Slide 123 / 179 Systems of linear equations can be solved using any of the three methods we previously discussed. efore solving a system, an analysis of the equations should be done to determine the "best" strategy to utilize. Graphing Choosing Strategy Slide 124 / 179 ltogether 292 tickets were sold for a basketball game. n 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? Substitution Elimination Step 1: Define your variables Let a = number of adult tickets Let s = number of student tickets Step 2: Set up the system Slide 125 / 179 Continued number of tickets sold: a + s = 292 money collected: 3a + s = 470 Slide 126 / 179 Continued a = 89 a + s = 292 89 + s = 292 s = 203 There were 89 adult tickets and 203 student tickets sold 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. Check: a + s = 292 89 + 203 = 292 292 = 292 3a + s = 470 3(89) + 203 = 470 267 + 203 = 470 470 = 470
Slide 127 / 179 44 What method would require the least amount of work to solve the following system: Slide 128 / 179 45 Solve the following system of linear equations using the method of your choice: y = 3x - 1 y = 3x - 1 y = 4x y = 4x (-4, -1) C graphing substitution elimination (-1, -4) C (-1, 4) D (1, 4) Slide 129 / 179 Slide 130 / 179 46 What method would require the least amount of work to solve the following system: 4s - 3t = 8 t = -2s -1 C graphing substitution elimination Slide 131 / 179 48 What method would require the least amount of work to solve the following system: 3m - 4n = 1 3m - 2n = -1 Slide 132 / 179 49 Solve the following system of linear equations using the method of your choice: 3m - 4n = 1 3m - 2n = -1 (-2, -1) C graphing substitution elimination (-1, -1) C (-1, 1) D (1, 1)
Slide 133 / 179 50 What method would require the least amount of work to solve the following system: Slide 134 / 179 51 Solve the following system of linear equations using the method of your choice: y = -x (-6, 12) graphing substitution (2, -4) Click for choices solved the system C (-2, 2) C elimination D (1, -2) Slide 135 / 179 Slide 136 / 179 52 What method would require the least amount of work to solve the following system: 53 Solve the following system of linear equations using the method of your choice: u = 4v 3u - 3v = 7 u = 4v 3u - 3v = 7 graphing substitution C (28, 7) C elimination D Slide 137 / 179 Slide 138 / 179 54 piece of glass with an initial temperature of 99 F is cooled at a rate of 3.5 F/min. t 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 55 Which method would you use to solve the system from the previous question? t = 99-3.5m t = 0 + 2.5m graphing 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 C substitution elimination
Slide 139 / 179 Slide 140 / 179 56 Solve the following system of linear equations: t = 99-3.5m Click to Reveal System t = 0 + 2.5m 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? m = 1 t = 2.5 m = 1 t = 95.5 C m = 16.5 t = 6.6 D m = 16.5 t = 41.25 1-1 C 3 D 4 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 141 / 179 Slide 142 / 179 Creating and Solving Systems Step 1: Define the variables Writing Systems to Model Situations Step 2: nalyze components and create equations Step 3: Solve the system utilizing the best strategy Return to Table of Contents Slide 143 / 179 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. total of 1,410 pounds of food was ordered. Slide 144 / 179 Continued Part : Using your work from part, find (1) the total number of adults in the group (2) the total number of children in the group Part : Write an equation or a system of equations that describe the above situation and define your variables. a + c = 148 12a + 9c = 1,410 a = number of adults c = number of children a + c = 148 12a + 9c = 1,410 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. (1) c = -a + 148 (2) 12a + 9(-a + 148) = 1410 12a - 9a + 1332 = 1410 3a = 78 a = 26 a + c = 148 26 + c = 148 c = 122
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 ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 146 / 179 Continued 3p + 2c = 6.00 2p + 3c = 5.25 Elimination: Multiply first equation by 2 Multiply second equation by -3 3p + 2c = 6.00 6p + 4c = 12 3p + 2(0.75) = 6-6p - 9c = -15.75 3p + 1.5 = 6-5c = -3.75 3p = 4.5 c = 0.75 p = 1.5 Cola: $0.75 Pizza: $1.50 Slide 147 / 179 58 Your class receives $1,105 for selling 205 packages of greeting cards and gift wrap. 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? Slide 148 / 179 59 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. 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? You will how many packages of gift wrap in the next question. Slide 149 / 179 Slide 150 / 179 60 The sum of two numbers is 47, and their difference is 15. What is the larger number? 16 31 C 32 D 36 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? From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011.
Slide 151 / 179 Slide 152 / 179 62 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many quarters do you have? 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 153 / 179 Slide 154 / 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? 65 Mary and my had a total of 20 yards of material from which to make costumes. Mary used three times more material to make her costume than my used, and 2 yards of material was not used. How many yards of material did my use for her costume? $0.50 $0.75 C $1.00 D $2.00 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 155 / 179 Slide 156 / 179 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? Solving Systems of Inequalities From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Return to Table of Contents
Slide 157 / 179 Vocabulary Slide 158 / 179 Graphing a System of Linear Inequalities 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. 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. Slide 159 / 179 Solve the following system of linear inequalities. y < -1x + 3 2 y < 1x Step 1: 4 Step 2: Slide 160 / 179 Continued y < -1x + 3 2 y < 1x 4 Step 3: Slide 161 / 179 Continued y < -1x + 3 2 y < 1x 4 Slide 162 / 179 Solve the following system of linear inequalities. Step 1: 2x + y > -4 x - 2y < 4
Step 2: Slide 163 / 179 Continued 2x + y > -4 x - 2y < 4 Step 3: Slide 164 / 179 Continued 2x + y > -4 x - 2y < 4 Slide 165 / 179 Solve the following system of linear inequalities. Step 1: 4x + 2y < 8 4x + 2y > -8 Step 2: Slide 166 / 179 Continued 4x + 2y < 8 4x + 2y > -8 Step 3: Slide 167 / 179 Continued 4x + 2y < 8 4x + 2y > -8 Slide 168 / 179 Solve the following system of linear inequalities. y < 3 x > 1 Step 1:
Slide 169 / 179 Continued y < 3 Slide 170 / 179 Continued y < 3 Step 2: x > 1 Step 3: x > 1 Slide 171 / 179 Slide 172 / 179 67 Choose the graph below that displays the solution to the following system of linear inequalities: 68 Choose the graph below that displays the solution to the following system of linear inequalities: y > -2x + 1 y < x + 2 x > 2 y < 5 C C Slide 173 / 179 Slide 174 / 179 69 Choose the graph below that displays the solution to the following system of linear inequalities: 70 Choose the graph below that displays the solution to the following system of linear inequalities: -5x + y > -2 4x + y < 1 3x + 2y < 12 2x - 2y < 20 C C
Slide 175 / 179 71 Choose all of the linear inequalities that correspond to the following graph: Slide 176 / 179 72 Which point is in the solution set of the system of inequalities shown in the accompanying graph? (0, 4) C (-4, 1) y > -2 C 3x + 4y > 12 (2, 4) D (4, -1) y < 2 D 3x + 4y < 12 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 177 / 179 Slide 178 / 179 73 Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph? 74 Which ordered pair is in the solution set of the following system of linear inequalities? y < 2x + 2 y x 1 (0, 3) (2, 0) (0, 0) (0, 1) C (1, 5) D (3, 2) C ( 1, 0) D ( 1, 4) From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 179 / 179 75 Mr. raun 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. raun can buy? From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011.