8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From our work in Section 6.1, we know that an equation of the form 3 is a linear equation. Remember that its graph is a straight line. Often we will want to consider two equations together. The then form a sstem of linear equations. An eample of such a sstem is 3 3 5 A solution for a linear equation in two variables is an ordered pair that satisfies the equation. Often there is just one ordered pair that satisfies both equations of a sstem. It is called the solution for the sstem. For instance, there is one solution for the sstem above, and it is (2, 1) because, replacing with 2 and with 1, we have 3 3 5 2 1 3 3 2 1 5 3 3 6 1 5 5 5 Because both statements are true, the ordered pair (2, 1) satisfies both equations. One approach to finding the solution for a sstem of linear equations is the graphical method. To use this, we graph the two lines on the same coordinate sstem. The coordinates of the point where the lines intersect is the solution for the sstem. Eample 1 Solving b Graphing NOTE Use the intercept method to graph each equation. Solve the sstem b graphing. 6 4 First, we determine solutions for the equations of our sstem. For 6, two solutions are (6, 0) and (0, 6). For 4, two solutions are (4, 0) and (0, 4). Using these intercepts, we graph the two equations. The lines intersect at the point (5, 1). NOTE B substituting 5 for and 1 for into the two original equations, we can check that (5, 1) is indeed the solution for our sstem. 6 4 5 1 6 5 1 4 6 6 4 4 Both statements must be true for (5, 1) to be a solution for the sstem. 4 (5, 1) 6 (5, 1) is the solution of the sstem. It is the onl point that lies on both lines. 621
622 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS CHECK YOURSELF 1 Solve the sstem b graphing. 2 4 5 Eample 2 shows how to graph a sstem when one of the equations represents a horizontal line. Eample 2 Solving b Graphing Solve the sstem b graphing. 3 2 6 6 For 3 2 6, two solutions are (2, 0) and (0, 3). These represent the and intercepts of the graph of the equation. The equation 6 represents a horizontal line that crosses the ais at the point (0, 6). Using these intercepts, we graph the two equations. The lines will intersect at the point ( 2, 6). So this is the solution to our sstem. 6 3 2 6
SYSTEMS OF LINEAR EQUATIONS: SOLVING BY GRAPHING SECTION 8.1 623 CHECK YOURSELF 2 Solve the sstem b graphing. 4 5 20 8 The sstems in Eamples 1 and 2 both had eactl one solution. A sstem with one solution is called a consistent sstem. It is possible that a sstem of equations will have no solution. Such a sstem is called an inconsistent sstem. We present such a sstem here. Eample 3 Solving an Inconsistent Sstem Solve b graphing. 2 2 2 4 We can graph the two lines as before. For 2 2, two solutions are (0, 2) and (1, 0). For 2 4, two solutions are (0, 4) and (2, 0). Using these intercepts, we graph the two equations. NOTE In slope-intercept form, our equations are 2 2 and 2 4 Both lines have slope 2. 2 2 2 4 Notice that the slope for each of these lines is 2, but the have different intercepts. This means that the lines are parallel (the will never intersect). Because the lines have no points in common, there is no ordered pair that will satisf both equations. The sstem has no solution. It is inconsistent.
624 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS CHECK YOURSELF 3 Solve b graphing. 3 3 3 6 There is one more possibilit for linear sstems, as Eample 4 illustrates. Eample 4 Solving a Dependent Sstem NOTE Notice that multipling the first equation b 2 results in the second equation. Solve b graphing. 2 4 2 4 8 Graphing as before and using the intercept method, we find The two equations have the same graph! 2 4 2 4 8 Because the graphs coincide, there are infinitel man solutions for this sstem. Ever point on the graph of 2 4 is also on the graph of 2 4 8, so an ordered pair satisfing 2 4 also satisfies 2 4 8. This is called a dependent sstem, and an point on the line is a solution.
SYSTEMS OF LINEAR EQUATIONS: SOLVING BY GRAPHING SECTION 8.1 625 CHECK YOURSELF 4 Solve b graphing. 4 2 2 8 The following summarizes our work in this section. Step b Step: To Solve a Sstem of Equations b Graphing Step 1 Step 2 Graph both equations on the same coordinate sstem. Determine the solution to the sstem as follows. a. If the lines intersect at one point, the solution is the ordered pair corresponding to that point. This is called a consistent sstem. A consistent sstem NOTE There is no ordered pair that lies on both lines. b. If the lines are parallel, there are no solutions. This is called an inconsistent sstem. An inconsistent sstem
626 CHAPTER 8 SYSTEMS OF LINEAR EQUATIONS NOTE An ordered pair that corresponds to a point on the line is a solution. c. If the two equations have the same graph, then the sstem has infinitel man solutions. This is called a dependent sstem. A dependent sstem Step 3 Check the solution in both equations, if necessar. CHECK YOURSELF ANSWERS 1. 2. 5 ( 5, 8) 8 (3, 2) 4 5 20 2 4 3. There is no solution. The lines are parallel, so the sstem is inconsistent. 3 3 3 6 4. 4 2 2 8 A dependent sstem
Name 8.1 Eercises Section Date Solve each of the following sstems b graphing. 1. 6 2. 8 4 2 ANSWERS 1. 2. 3. 4. 5. 6. 3. 3 4. 7 5 3 5. 2 4 6. 3 6 1 4 627
ANSWERS 7. 8. 9. 7. 2 8 8. 2 2 2 0 2 6 10. 11. 12. 9. 3 12 10. 2 4 2 3 6 2 6 11. 3 2 12 12. 2 8 3 3 2 12 628
ANSWERS 13. 4 14. 2 8 2 2 8 2 13. 14. 15. 16. 17. 18. 15. 4 4 16. 6 6 2 8 4 17. 3 2 6 18. 4 3 12 2 5 2 629
ANSWERS 19. 20. 21. 19. 3 3 20. 3 6 9 3 6 2 3 22. 23. 24. 21. 2 3 22. 6 2 3 2 6 23. 4 24. 3 6 5 630
ANSWERS 25. Find values for m and b in the following sstem so that the solution to the sstem is (1, 2). 25. 26. m 3 8 3 4 b 27. 28. 26. Find values for m and b in the following sstem so that the solution to the sstem is ( 3, 4). 5 7 b m 22 29. 30. 27. Complete the following statements in our own words: To solve an equation means to.... To solve a sstem of equations means to.... 28. A sstem of equations such as the one below is sometimes called a 2-b-2 sstem of linear equations. 3 4 1 2 6 Eplain this term. 29. Complete this statement in our own words: All the points on the graph of the equation 2 3 6.... Echange statements with other students. Do ou agree with other students statements? 30. Does a sstem of linear equations alwas have a solution? How can ou tell without graphing that a sstem of two equations graphs into two parallel lines? Give some eamples to eplain our reasoning. 631
ANSWERS a. b. c. d. e. f. Getting Read for Section 8.2 [Section 1.6] Simplif each of the following epressions. (a) (2 ) ( ) (b) ( ) ( ) (c) (3 2) ( 3 3) (d) ( 5) (2 5) (e) 2( ) (3 2) (f ) 2(2 ) ( 4 3) (g) 3(2 ) 2( 3 ) (h) 3(2 4) 4( 3) g. h. Answers 1. 6 3. (5, 1) 4 3 5 (1, 4) 5. 2 4 7. (2, 1) 1 2 8 2 0 (2, 4) 632
9. 3 12 11. (6, 2) 2 3 6 3 2 12 3 (2, 3) 13. 4 15. Dependent 2 2 8 4 4 2 8 (4, 2) 17. 3 2 6 19. (4, 3) 2 5 3 3 3 6 Inconsistent 633
21. 2 3 23. 2 3 0, 3 2 4 6 (4, 6) 25. m 2, b 5 27. 29. a. 3 b. 2 c. d. 3 e. 5 f. 5 g. 5 h. 10 634