Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

3.1 Systems of Equations in Two Variables ■ Translating ■ Identifying Solutions ■ Solving Systems Graphically ■ Models

Slide 3- 3 Copyright © 2012 Pearson Education, Inc. Solutions of Systems A system of equations is a set of two or more equations that are to be solved simultaneously. A solution of a system of two equations in two variables is an ordered pair of numbers that makes both equations true. The numbers in the ordered pair correspond to the variables in alphabetical order.

Slide 3- 4 Copyright © 2012 Pearson Education, Inc. 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) = 7 TRUE The pair (1, 5) makes both equations true, so it is a solution of the system.

Slide 3- 5 Copyright © 2012 Pearson Education, Inc. Solving Systems of Equations by Graphing Recall that a graph of an equation is a set of points representing its solution set. Each point on the graph corresponds to an ordered pair that is a solution of the equation. By graphing two equations using one set of axes, we can identify a solution of both equations by looking for a point of intersection.

Slide 3- 6 Copyright © 2012 Pearson Education, Inc. Example Solve this system of equations by graphing: y = 3x + 1 x  2y = 3 Solution Graph each equation: y = 3x + 1 Graph (0, 1) and “count” off a slope of 3. Next graph: x  2y = 3 Graph using the intercepts. (0,  3/2) (3, 0) (  1,  2) The common point gives the common solution.

Slide 3- 7 Copyright © 2012 Pearson Education, Inc. Example—graphing calculator Solve this system of equations by graphing: y = 3x + 1 x  2y = 3 Solution Solve each equation for y. y = 3x + 1 x – 2y = 3 – 2y = 3 – x y = – 3/2 + x/2 Enter both equations in the same viewing window.

Slide 3- 8 Copyright © 2012 Pearson Education, Inc. Example Solve this system of equations by graphing: Solution Graph each equation. Both equations are in slope- intercept form so it is easy to see that both lines have the same slope. The y-intercepts differ so the lines are parallel. Because the lines are parallel, there is no point of intersection. The system has no solution. y = 3x/4 + 2 y = 3x/4  3

Slide 3- 9 Copyright © 2012 Pearson Education, Inc. Example Solve this system of equations by graphing: Solution Graph the equations. Both equations represent the same line. Because the equations are equivalent, any solution of one equation is a solution of the other equation as well. 4x + 8y = 16 2x + 4y = 8 (0, 2) (2, 1)

Slide Copyright © 2012 Pearson Education, Inc. Classifying Systems of Equations 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. Graphs intersect at one point. The system is consistent and has one solution. Independent equations: Since neither equation is a multiple of the other, they are independent. Equations have the same graph. The system is consistent and has an infinite number of solutions. The equations are dependent since they are equivalent. Graphs are parallel. The system is inconsistent because there is no solution. Since the equations are not equivalent, they are independent.