1. Holt 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

2. Holt 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.

3. Holt 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.

4. Holt 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

5. Holt 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

6. Holt 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.

7. Holt Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Use a graph 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

8. Holt 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

9. Holt Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Use a graph 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 =

10. Holt 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

11. Holt 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.

12. Holt 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.

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

14. Holt 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

15. Holt 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.

16. Holt Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Check A graph shows parallel lines. Example 3B Continued

17. Holt 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

18. Holt 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.

19. Holt Algebra 2 3-1 Using Graphs and Tables to Solve Linear Systems Step 2 Solve the system by using a calculator. xy 020 47.5 175 102.5 2120 xy 035 57.5 180 102.5 2125 y = 55x + 20y = 45x + 35 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 You can use a table to check your answer.