1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Systems of Equations in Two Variables Translating Identifying Solutions Solving Systems Graphically 3.1

2. Slide 3- 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley System of Equations A system of equations is a set of two or more equations, in two or more variables, for which a common solution is sought.

3. Slide 3- 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution T-shirt Villa sold 52 shirts, one kind at $8.25 and another at $11.50 each. In all, $464.75 was taken in for the shirts. How many of each kind were sold? 1. Familiarize. To familiarize ourselves with this problem, guess that 26 of each kind of shirt was sold. The total money taken in would be The guess is incorrect, now turn to algebra.

4. Slide 3- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2. Translate. Let x = the number of $8.25 shirts and y = the number of $11.50 shirts. We have the following system of equations: x + y = 52 8.25x + 11.50y = 464.75 Kind of Shirt $8.25 shirt $11.50 shirt Total Number sold xy52 Price$8.25$11.50 Amount$8.25x$11.50y$464.75

5. Slide 3- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Identifying Solutions A solution of a system of two equations in two variables is an ordered pair of numbers that makes both equations true.

6. Slide 3- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Determine whether (1, 5) is a solution of the system x – y = – 4 1 – 5 – 4 – 4 = – 4 TRUE 2x + y = 7 2(1) + 5 7 7 = 7 TRUE The pair (1, 5) makes both equations true, so it is a solution of the system.

7. Slide 3- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Systems Graphically One way to solve a system of two equations is to graph both equations and identify any points of intersection. The coordinates of each point of intersection represent a solution of that system.

8. Slide 3- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve the system graphically. Solution It appears that (3, 2) is the solution. A check by substituting into both equations shows that (3, 2) is indeed the solution. x y -5 -4 -3 -2 -1 1 2 3 4 5 4 -2 -4 -3 3 2 5 1 6 x – y = 1 7 x + y = 5 (3, 2)

9. Slide 3- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve the system graphically. Solution x y -5 -4 -3 -2 -1 1 2 3 4 5 4 -2 -4 -3 3 2 5 1 6 y = 2x – 1 7 y = 2x + 3 The lines have the same slope and different y-intercepts, so they are parallel. The system has no solution.

10. Slide 3- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve the system graphically. Solution The same line is drawn twice. Any solution of one equation is a solution of the other. There is an infinite number of solutions. The solution set is x y -5 -4 -3 -2 -1 1 2 3 4 5 4 -2 -4 -3 3 2 5 1 6 2y – 6x = 12 7 3x – y = –6

11. Slide 3- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When we graph a system of two linear equations in two variables, one of the following three outcomes will occur. 1.The lines have one point in common, and that point is the only solution of the system. Any system that has at least one solution is said to be consistent. 2.The lines are parallel, with no point in common, and the system has no solution. This type of system is called inconsistent. 3.The lines coincide, sharing the same graph. This type of system has an infinite number of solutions and is also said to be consistent.

12. Slide 3- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When one equation in a system can be obtained by multiplying both sides of another equation by a constant, the two equations are said to be dependent. If two equations are not dependent, they are said to be independent.