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Section 3.2 Linear Equations in Two Variables Copyright © 2013 Pearson Education, Inc. Example Page 169 Determine whether the given ordered pair is a solution to the given equation. a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4) Solution a. y=x+5 7=2+5 7 = 7 True The ordered pair (2, 7) is a solution. Example Page 169 Determine whether the given ordered pair is a solution to the given equation. a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4) Solution b. 2x + 3y = 18 2(3) + 3(4) = 18 6 12 = 18 6 18 The ordered pair (3, 4) is NOT a solution. Example Page 170 A table can be used to list solutions to an equation. Complete the table for the equation y = 3x – 1. x y 3 1 0 10 4 1 3 8 Solution x 3 x 1 x0 x 3 y 3x 1 y 3( 3) 1 y 9 1 y 3x 1 y 3( 1) 1 y 3 1 y 3x 1 y 3x 1 y 3(0) 1 y 0 1 y 3(3) 1 y 9 1 y 10 y 4 y 1 y 8 Finding Solutions of an Equation Page 170 A table that lists a few solutions is helpful when graphing an equation y = 3x + 2. Find five solutions to the equation Choose some x values and then compute the corresponding y values and complete the table. X Y (x,y) If x = -2, y = 3(-2) + 2 = -4. Ordered pair -2 4 (2,4) If x = -1, y = 3(-1) + 2 = -1. Ordered pair -1 1 (1,1) If x =0, y = 3(0) + 2 = 2. Ordered pair 0 2 (0,2) If x =1, y = 3(1) + 2 = 5. Ordered pair 1 5 (1,5) If x =2, y = 3(2) + 2 = 8. Ordered pair 2 8 ( 2,8) Graph on next slide. Graph of the Equation Plot the five ordered pairs to obtain the graph of y = 3x + 2 X Y (X,Y) -2 -4 (-2,-4) -1 -1 (-1,-1) 0 2 (0,2) 1 5 (1,5) 2 8 (2,8) Example Page 171 Make a table of values for the equation y = 3x, and then use the table to graph this equation. Solution Start by selecting a few convenient values for x such as –1, 0, 1, and 2. Then complete the table. x y –1 –3 0 0 1 3 2 6 Plot the points and connect the points with a straight line. Linear Equation in Two Variables Page 172 A linear equation in two variables can be written as Ax + By = C, where A, B, and C are fixed numbers (constants) and A and B are not both equal to 0. The graph of a linear equation in two variables is a line. Example Graph the linear equation. y 1 x 1 4 Solution Because this equation can be written in standard form, it is a linear equation. Choose any three values for x. x –4 y 0 0 4 1 2 Plot the points and connect the points with a straight line. Page 173 Example Page 173 Graph the linear equation. x y 5 Solution Because this equation can be written in standard form, it is a linear equation. Choose any three values for x. x 0 2 5 y 5 3 0 Plot the points and connect the points with a straight line. Example Page 174 Graph the linear equation by solving for y first. 3x 6 y 12 Solution Solve for y. 3 x 6 y 12 6 y 3 x 12 1 y x2 2 x –2 0 2 y 1 2 3 Graph of the Equation Plot the five ordered pairs to obtain the graph of y = 3x + 1 (2,7) (1,4) (0,1) (-1,-2) (-2,-5) X Y (x,y) 2 7 (2,7) 1 4 (1,4) 0 1 (0,1) -1 -2 (-1,-2) -2 -5 (-2,-5) DONE Tables of Solutions Page 170 A table can be used to list solutions to an equation. A table that lists a few solutions is helpful when graphing an equation. Basic Concepts Page 168 Equations can have any number of variables. A solution to an equation with one variable is one number that makes the statement true. Page 172 Example Page 174 Graph the linear equation by solving for y first. 3x 6 y 12 Solution Solve for y. 3 x 6 y 12 6 y 3 x 12 1 y x2 2 x –2 0 2 y 1 2 3 Graph of the Equation Plot the ordered pairs to obtain the graph of y = 2x y 2(1) y2 y 2 (0) y0 y 2 (2) y4 y 2 (2) y 4 Graph of the Equation Plot the ordered pairs to obtain the graph of y = 2x-2 y 2(1) 2 y0 y 2 ( 0) 2 y 2 y 2 ( 2) 2 y2 y 2 (2) 2 y 6 Graph of the Equation Plot the ordered pairs to obtain the graph of 1 y x2 2 1 ( 2) 2 2 y3 y 1 (0) 2 2 y2 y 1 (4) 2 2 y4 y 1 ( 2) 2 2 y 1 y Graph of the Equation Plot the five ordered pairs to obtain the graph of y = 3x + 2 X Y (x,y) -2 -4 (-2,-4) -1 -1 (-1,-1) 0 2 (0,2) 1 5 (1,5) 2 8 (2,8)