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Holt Algebra Using Graphs and Tables to Solve Linear Systems 3-1 Using Graphs and Tables to Solve Linear Systems Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Warm Up Use substitution to determine if (1, –2) is an element of the solution set of the linear equation. 1. y = 2x y = 3x – 5 no yes Write each equation in slope-intercept form. 3. 2y + 8x = y – 3x = 8 y = –4x + 3

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Solve systems of equations by using graphs and tables. Determine the number of solutions of a system of equations. Objectives

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Holt Algebra Using Graphs and Tables to Solve Linear Systems system of equations linear system Vocabulary

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Holt Algebra Using Graphs and Tables to Solve Linear Systems A system of equations is a set of two or more equations containing two or more variables. A linear system is a system of equations containing only linear equations. Recall that a line is an infinite set of points that are solutions to a linear equation. The solution of a system of equations is the set of all points that satisfy each equation.

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Holt Algebra Using Graphs and Tables to Solve Linear Systems On the graph of the system of two equations, the solution is the set of points where the lines intersect. A point is a solution to a system of equation if the x- and y-values of the point satisfy both equations.

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations. Example 1A: Verifying Solutions of Linear Systems (1, 3); x – 3y = –8 3x + 2y = 9 Substitute 1 for x and 3 for y in each equation. Because the point is a solution for both equations, it is a solution of the system. 3x + 2y = 9 3(1) +2(3) x – 3y = –8 (1) –3(3) –8–8 –8–8–8–8

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations. Example 1B: Verifying Solutions of Linear Systems (–4, ); x + 6 = 4y 2x + 8y = 1 x + 6 = 4y (–4) Because the point is not a solution for both equations, it is not a solution of the system. 2x + 8y = 1 2(–4) + 1 –4–4 1 Substitute –4 for x and for y in each equation. x

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Check It Out! Example 1a Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations. (4, 3); x + 2y = 10 3x – y = 9 Because the point is a solution for both equations, it is a solution of the system Substitute 4 for x and 3 for y in each equation. x + 2y = 10 (4) + 2(3) (4) – (3) 3x – y = 9 9 9

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Check It Out! Example 1b Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations. (5, 3); 6x – 7y = 1 3x + 7y = 5 Substitute 5 for x and 3 for y in each equation. Because the point is not a solution for both equations, it is not a solution of the system. 5 3(5) + 7(3) 3x + 7y = x 6x – 7y = 1 6(5) – 7(3) x

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Recall that you can use graphs or tables to find some of the solutions to a linear equation. You can do the same to find solutions to linear systems.

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Use a graph and a table to solve the system. Check your answer. Example 2A: Solving Linear Systems by Using Graphs and Tables 2x – 3y = 3 y + 2 = x Solve each equation for y. y= x – 2 y= x – 1

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Holt Algebra Using Graphs and Tables to Solve Linear Systems On the graph, the lines appear to intersect at the ordered pair (3, 1) Example 2A Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Make a table of values for each equation. Notice that when x = 3, the y-value for both equations is –1–10 yx xy 0 –2–2 1– y= x – 2 y= x – 1 The solution to the system is (3, 1). Example 2A Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Use a graph and a table to solve the system. Check your answer. Example 2B: Solving Linear Systems by Using Graphs and Tables x – y = 2 2y – 3x = –1 Solve each equation for y. y = x – 2 y =

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Example 2B Continued Use your graphing calculator to graph the equations and make a table of values. The lines appear to intersect at (–3, –5). This is the confirmed by the tables of values. Check Substitute (–3, –5) in the original equations to verify the solution. The solution to the system is (–3, –5). 2(–5) – 3(–3) 2y – 3x = –1 –1 (–3) – (–5) 2 x – y = 2 2 2

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Use a graph and a table to solve the system. Check your answer. 2y + 6 = x 4x = 3 + y Check It Out! Example 2a Solve each equation for y. y= 4x – 3 y= x – 3

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Holt Algebra Using Graphs and Tables to Solve Linear Systems On the graph, the lines appear to intersect at the ordered pair (0, –3) Check It Out! Example 2a Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Make a table of values for each equation. Notice that when x = 0, the y-value for both equations is –3. 3 –22 1 –3–3 0 yx xy 0 –3– The solution to the system is (0, –3). y = 4x – 3y = x – 3 Check It Out! Example 2a Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Use a graph and a table to solve the system. Check your answer. x + y = 8 2x – y = 4 Check It Out! Example 2b y = 2x – 4 y = 8 – x Solve each equation for y.

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Holt Algebra Using Graphs and Tables to Solve Linear Systems On the graph, the lines appear to intersect at the ordered pair (4, 4). Check It Out! Example 2b Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Make a table of values for each equation. Notice that when x = 4, the y-value for both equations is 4. xy 1 –2– xy y = 2x – 4y= 8 – x The solution to the system is (4, 4). Check It Out! Example 2b Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Use a graph and a table to solve each system. Check your answer. y – x = 5 3x + y = 1 y= –3x + 1 y= x + 5 Check It Out! Example 2c Solve each equation for y.

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Holt Algebra Using Graphs and Tables to Solve Linear Systems On the graph, the lines appear to intersect at the ordered pair (–1, 4). Check It Out! Example 2c Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Make a table of values for each equation. Notice that when x = –1, the y-value for both equations is 4. xy –1– –2–2 2–5–5 xy –1– The solution to the system is (–1, 4). y= –3x + 1y= x + 5 Check It Out! Example 2c Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems The systems of equations in Example 2 have exactly one solution. However, linear systems may also have infinitely many or no solutions.

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Holt Algebra Using Graphs and Tables to Solve Linear Systems

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Determine the number of solutions. Example 3A: Classifying Linear System x = 2y + 6 3x – 6y = 18 Solve each equation for y. The system has infinitely many solutions. The equations have the same slope and y-intercept and are graphed as the same line. y = x – 3

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Determine the number of solutions. Example 3B: Classifying Linear System 4x + y = 1 y + 1 = –4x Solve each equation for y. The system has no solution. y = –4x + 1 y = –4x – 1 The equations have the same slope but different y-intercepts and are graphed as parallel lines.

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Check A graph shows parallel lines. Example 3B Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Determine the number of solutions. 7x – y = –11 3y = 21x + 33 Solve each equation for y. The system has infinitely many solutions. The equations have the same slope and y-intercept and are graphed as the same line. Check It Out! Example 3a y = 7x + 11

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Determine the number of solutions. x + 4 = y 5y = 5x + 35 Solve each equation for y. The system has no solutions. The equations have the same slope but different y-intercepts and are graphed as parallel lines. Check It Out! Example 3b y = x + 4 y = x + 7

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Holt Algebra Using Graphs and Tables to Solve Linear Systems City Park Golf Course charges $20 to rent golf clubs plus $55 per hour for golf cart rental. Sea Vista Golf Course charges $35 to rent clubs plus $45 per hour to rent a cart. For what number of hours is the cost of renting clubs and a cart the same for each course? Example 4: Summer Sports Application

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Example 4 Continued Let x represent the number of hours and y represent the total cost in dollars. City Park Golf Course: y = 55x + 20 Sea Vista Golf Course: y = 45x + 35 The system has exactly one solution. Step 1 Write an equation for the cost of renting clubs and a cart at each golf course.

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Step 2 Solve the system by using a table of values. xy xy y = 55x + 20y = 45x + 35 Use increments of to represent 30 min. When x =, the y- values are both The cost of renting clubs and renting a cart for hours is $ at either company. So the cost is the same at each golf course for hours. Example 4 Continued

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Check It Out! Example 4 Ravi is comparing the costs of long distance calling cards. To use card A, it costs $0.50 to connect and then $0.05 per minute. To use card B, it costs $0.20 to connect and then $0.08 per minute. For what number of minutes does it cost the same amount to use each card for a single call? Step 1 Write an equation for the cost for each of the different long distance calling cards. Let x represent the number of minutes and y represent the total cost in dollars. Card A: y = 0.05x Card B: y = 0.08x

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Holt Algebra Using Graphs and Tables to Solve Linear Systems Check It Out! Example 4 Continued xy xy y = 0.05x y = 0.08x Step 2 Solve the system by using a table of values. When x = 10, the y- values are both The cost of using the phone cards of 10 minutes is $1.00 for either cards. So the cost is the same for each phone card at 10 minutes.

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