1. Copyright © Cengage Learning. All rights reserved. Systems of Linear Equations and Inequalities in Two Variables 7

2. Copyright © Cengage Learning. All rights reserved. Section 7.1 Solving Systems of Linear Equations by Graphing

3. 3 Objectives Determine whether an ordered pair is a solution to a given system of linear equations. Solve a system of linear equations by graphing. Recognize that an inconsistent system has no solution. Express the infinitely many solutions of a dependent system as a general ordered pair. 2 2 3 3 1 1 4 4

4. 4 Solving Systems of Linear Equations by Graphing The lines graphed below approximate the percentages of American households with only a landline phone and those with only a cell phone for the years 2005 to 2010. We can see that over this period, the percentage of those with only a landline decreased while those with only a cell phone increased.

5. 5 Solving Systems of Linear Equations by Graphing By graphing this information on the same coordinate system, it appears that the percentage of households with a cell phone only was the same as those with only a landline in October 2008—about 19.3% each. In this section, we will work with pairs of linear equations whose graphs often will be intersecting lines.

6. 6 Determine whether an ordered pair is a solution to a given system of linear equations 1.

7. 7 Determine whether an ordered pair is a solution to a given system of linear equations Recall that we have previously graphed equations such as x + y = 3 that contain two variables. Because there are infinitely many pairs of numbers whose sum is 3, there are infinitely many pairs (x, y) that will satisfy this equation. Some of these pairs are listed in Table 7-1(a). Table 7-1 (a)

8. 8 Determine whether an ordered pair is a solution to a given system of linear equations Likewise, there are infinitely many pairs (x, y) that will satisfy the equation 3x – y = 1. Some of these pairs are listed in Table 7-1(b). Although there are infinitely many pairs that satisfy each of these equations, only the pair (1, 2) satisfies both equations. Table 7-1 (b)

9. 9 Determine whether an ordered pair is a solution to a given system of linear equations The pair of equations x + y = 3 3x – y = 1 is called a system of equations. Because the ordered pair (1, 2) satisfies both equations, it is called a simultaneous solution or just a solution of the system of equations. We will discuss three methods for finding the solution of a system of two linear equations. In this section, we consider the graphing method.

10. 10 Solve a system of linear equations by graphing 2.

11. 11 Solve a system of linear equations by graphing To use the method of graphing to solve the system x + y = 3 3x – y = 1 we will graph both equations on one set of coordinate axes. Using the intercept method, Recall that to find the y-intercept, we let x = 0 and solve for y and to find the x-intercept, we let y = 0 and solve for x.

12. 12 Solve a system of linear equations by graphing We will also plot one extra point as a check. See Figure 7-2. Figure 7-2

13. 13 Solve a system of linear equations by graphing Although there are infinitely many pairs (x, y) that satisfy x + y = 3 and infinitely pairs (x, y) that satisfy 3x – y = 1, only the coordinates of the point where their graphs intersect satisfy both equations. The solution of the system is the ordered pair (1, 2). To check the solution, we substitute 1 for x and 2 for y in each equation and verify that the pair (1, 2) satisfies each equation.

14. 14 Solve a system of linear equations by graphing First equation Second equation x + y = 3 3x – y = 1 1 + 2 ≟ 3 3(1) – 2 ≟ 1 3 = 3 3 – 2 ≟ 1 1 = 1 When the graphs of two equations in a system are different lines, the equations are called independent equations. When a system of equations has a solution, the system is called a consistent system.

15. 15 Solve a system of linear equations by graphing To solve a system of equations in two variables by graphing, we follow these steps. The Graphing Method 1. Graph each equation on one set of coordinate axes. 2. Find the coordinates of the point where the graphs intersect, if applicable. 3. Check the solution in the equations of the original system, if applicable.

16. 16 Example 2x + 3y = 2 3x = 2y + 16 Solution: Using the intercept method, we graph both equations on one set of coordinate axes, as shown in Figure 7-3. We also plot a third point as a check. Figure 7-3 Solve the system.

17. 17 Example – Solution Although there are infinitely many pairs (x, y) that satisfy 2x + 3y = 2 and infinitely many pairs (x, y) that satisfy 3x = 2y + 16, only the coordinates of the point where the graphs intersect satisfy both equations. The solution is the ordered pair (4, –2). To check, we substitute 4 for x and –2 for y in each equation and verify that the pair (4, –2) satisfies each equation. 2x + 3y = 2 3x = 2y + 16 2(4) + 3(–2) ≟ 2 3(4) ≟ 2(–2) + 16 cont’d

18. 18 Example – Solution 8 – 6 ≟ 2 12 ≟ – 4 + 16 2 = 2 12 = 12 The equations in this system are independent equations, and the system is a consistent system of equations. cont’d

19. 19 Solve a system of linear equations by graphing Comment Always check your answer in the original equations. If you made an error in simplifying one of the equations, your answer would check in the simplified equations but not in the original.

20. 20 Recognize that an inconsistent system has no solution 3.

21. 21 Recognize that an inconsistent system has no solution Sometimes a system of equations will have no solution. These systems are called inconsistent systems.

22. 22 Example 2x + y = –6 4x + 2y = 8 Solution: We graph both equations on one set of coordinate axes, as in Figure 7-5. Figure 7-5 Solve the system by graphing:.

23. 23 Example – Solution Since the graphs are different lines, the equations of the system are independent. The lines in the figure appear to be parallel. To be sure, we can find their slopes. Recall that if two lines have the same slope but different y-intercepts, they will be parallel. To determine the slope of each line, we write each equation in slope-intercept form, y = mx + b. 2x + y = –6 4x + 2y = 8 y = –2x – 62y = –4x + 8 y = –2x + 4 cont’d

24. 24 Example – Solution Because both equations have the same slope (–2) but different y-intercepts (0, –6) and (0, 4), they are parallel. Since parallel lines do not intersect, the system is inconsistent. Its solution set is ∅. cont’d

25. 25 Express the infinitely many solutions of a dependent system as a general ordered pair 4.

26. 26 Express the infinitely many solutions of a dependent system as a general ordered pair Sometimes a system will have infinitely many solutions. In this case, we say that the equations of the system are dependent equations.

27. 27 Example y – 2x = 4 4x + 8 = 2y Solution: We graph each equation on one set of axes, as in Figure 7-6. Figure 7-6 Solve the system by graphing:.

28. 28 Example – Solution The lines in the figure appear to be the same line. To be sure, we can find their slopes and y-intercepts by writing each equation in slope-intercept form. If two lines have the same slope and the same y-intercept, they will be the same line. y – 2x = 44x + 8 = 2y y = 2x + 42x + 4 = y cont’d

29. 29 Example – Solution Since the lines in the figure are the same line, they intersect at infinitely many points and there are infinitely many solutions. Every solution to the first equation is a solution to the second equation. To describe these solutions, we can solve either equation for y. Because 2x + 4 is equal to y, every solution (x, y) of the system will have the form (x, 2x + 4). This solution can also be written in set-builder notation, {(x, y) | y = 2x + 4}. cont’d

30. 30 Example – Solution To find some specific solutions, we can substitute 0, –3, and –1 for x in the general ordered pair (x, 2x + 4) to obtain (0, 4), (–3, –2), and (–1, 2). From the graph, we can see that each point lies on the one line that is the graph of both equations. cont’d

31. 31 Table 7-2 summarizes the possibilities that can occur when two nonvertical linear equations, each with two variables, are graphed. Table 7-2 Express the infinitely many solutions of a dependent system as a general ordered pair