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Holt McDougal Algebra 2 3-1 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 Holt McDougal Algebra 2

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3-1 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 + 12. y = 3x – 5 no yes Write each equation in slope-intercept form. 3. 2y + 8x = 6 4. 4y – 3x = 8 y = –4x + 3

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Holt McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Solve systems of equations by using graphs and tables. Classify systems of equations, and determine the number of solutions. Objectives

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Holt McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems system of equations linear system consistent system inconsistent system independent system dependent system Vocabulary

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Holt McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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) 9 9 9 x – 3y = –8 (1) –3(3) –8–8 –8–8–8–8

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Holt McDougal Algebra 2 3-1 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) + 6 22 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 McDougal Algebra 2 3-1 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) 10 9 3(4) – (3) 3x – y = 9 9 9

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Holt McDougal Algebra 2 3-1 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 = 5 5 36 x 6x – 7y = 1 6(5) – 7(3) 1 9 1 x

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Holt McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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. 13 2 1 –1–10 yxxy 0 –2–2 1– 1 20 31 y= x – 2 y= x – 1 The solution to the system is (3, 1). Example 2A Continued

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Holt McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 yxxy 0 –3–3 11 25 39 The solution to the system is (0, –3). y = 4x – 3y = x – 3 Check It Out! Example 2a Continued

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Holt McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 –2–2 20 32 44 xy 1 7 26 35 44 y = 2x – 4y= 8 – x The solution to the system is (4, 4). Check It Out! Example 2b Continued

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Holt McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 –1–1 4 01 1–2–2 2–5–5 xy –1–1 4 05 16 27 The solution to the system is (–1, 4). y= –3x + 1y= x + 5 Check It Out! Example 2c Continued

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Holt McDougal Algebra 2 3-1 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. A consistent system is a set of equations or inequalities that has at least one solution, and an inconsistent system will have no solutions.

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Holt McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems You can classify linear systems by comparing the slopes and y-intercepts of the equations. An independent system has equations with different slopes. A dependent system has equations with equal slopes and equal y-intercepts.

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Holt McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems

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Holt McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Classify the system and determine the number of solutions. Example 3A: Classifying Linear System x = 2y + 6 3x – 6y = 18 Solve each equation for y. The system is consistent and dependent with 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 McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Classify the system and determine the number of solutions. Example 3B: Classifying Linear System 4x + y = 1 y + 1 = –4x Solve each equation for y. The system is inconsistent and 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 McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Check A graph shows parallel lines. Example 3B Continued

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Holt McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Classify the system and determine the number of solutions. 7x – y = –11 3y = 21x + 33 Solve each equation for y. The system is consistent and dependent with 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 McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Classify each system and determine the number of solutions. x + 4 = y 5y = 5x + 35 Solve each equation for y. The system is inconsistent with no solution. 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 McDougal Algebra 2 3-1 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 McDougal Algebra 2 3-1 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 Because the slopes are different, the system is independent and 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 McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Step 2 Solve the system by using a table of values. xy 020 47.5 175 102.5 2120 xy 035 57.5 180 102.5 2125 y = 55x + 20y = 45x + 35 Use increments of to represent 30 min. When x =, the y- values are both 102.5. The cost of renting clubs and renting a cart for hours is $102.50 at either company. So the cost is the same at each golf course for hours. Example 4 Continued

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Holt McDougal Algebra 2 3-1 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 + 0.50 Card B: y = 0.08x + 0.20

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Holt McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Check It Out! Example 4 Continued xy 1 0.28 50.60 101.00 151.40 xy 1 0.55 50.75 101.00 151.25 y = 0.05x + 0.50y = 0.08x + 0.20 Step 2 Solve the system by using a table of values. When x = 10, the y- values are both 1.00. 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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Holt McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Lesson Quiz: Part I Use substitution to determine if the given ordered pair is an element of the solution set of the system of equations. x + y = 2 y + 2x = 5 x + 3y = –9 y – 2x = 4 no yes 1. (4, –2) 2. (–3, –2) Solve the system using a table and graph. Check your answer. 3. x + y = 1 3x –2y = 8 (2, –1)

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Holt McDougal Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Lesson Quiz: Part II Classify each system and determine the number of solutions. 4. 5. –4x = 2y – 10 y + 2x = –10 inconsistent; none consistent, dependent; infinitely many 6. Kayak Kottage charges $26 to rent a kayak plus $24 per hour for lessons. Power Paddles charges $12 for rental plus $32 per hour for lessons. For what number of hours is the cost of equipment and lessons the same for each company?

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