Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Section 4.1 Solving Systems of Equations in Two Variables by Graphing

2 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Systems of Equations A system of equations or system of inequalities is two or more equations or inequalities in several variables that are considered simultaneously. x y 11 22 33 123 22 3 The lines may intersect. x y 11 22 33 123 22 3 The lines may be parallel. x y 11 22 33 123 22 3 The lines may coincide.

3 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. xy 22 0 0  6 1  9 Solve by graphing. y =  3x  6 y = 2x  1 xy 22  5 0  1 23 x y 22 44 66 8 44 66 8 Continued Example

4 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. x y 22 44 66 8 44 66 8 The lines intersect at (  1,  3). Check: y =  3x  6 y = 2x  1  3 =  3(  1)  6  3 = 3  6  3 =  3  3 = 2(  1)  1  3 =  2  1  3 =  3 y =  3x  6 y = 2x  1 A system of equations that has one solution is said to be consistent. The solution of the system is (–1, –3). Example (cont)

5 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Solve by graphing. 3x  2y =  4  9x + 6y =  6 x y 22 44 66 8 44 66 8 x  2y =  4  9x + 6y =  6 A system of linear equations that has no solution is called an inconsistent system. The lines are parallel. The lines do not intersect, so there is no solution. Example

6 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Solve by graphing. 4x  6y = 8  2x + 3y =  4 x y 22 44 66 8 44 66 8 x  6y = 8  2x + 3y =  4 A system of linear equations that has an infinite number of solutions is called a dependent system. The lines coincide. Every point on each graph coincides, thus there are an infinite number of solutions. Example

7 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. Walter and Barbara need some plumbing repairs done at their house. They called two companies for estimates of the work that needs to be done. Roberts Plumbing and Heating charges $40 for a house call and then $35 per four for labor. Instant Plumbing Repairs charges $70 for a house call and then $25 per hour for labor. a. Create a cost equation for each company, where y is the total cost of plumbing repairs and x is the number of hours of labor. Write the system of equations. b. Graph the two equations using the values x = 0.3 and 6. c. Determine from your graph how many hours of plumbing repairs would be required for the two companies to charge the same. d. Determine from your graph which company charges less if the estimated amount of time is 4 hours. Example Continued

8 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. a. Create a cost equation for each company, where y is the total cost of plumbing repairs and x is the number of hours of labor. Write the system of equations. Example (cont) Continued Total cost of plumbing = Cost of house call cost per hour +× number of labor hours y = × x y = × x y = x y = x

9 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. b. Graph. Example (cont) Continued xy Roberts Plumbing and Heating Instant Plumbing Repairs xy y = x y = x

10 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. c. Determine from your graph how many hours of plumbing repairs would be required for the two companies to charge the same. Example (cont) Continued (3, 145) The lines intersect at (3, 145), Thus the two companies will charge the same if 3 hours of plumbing repairs are required.

11 Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. d. Determine from your graph which company charges less if the estimated amount of time is 4 hours. Example (cont) We draw a dashed line at x = 4. We see that the blue line is higher than the red line. Thus, the cost would be less if Walter and Barbara use Instant Plumbing Repairs for 4 hours of work.