11.2 graphing linear equations in two variables

11.2 Graphing Linear Equations in Two Variables

1. 2 Graphs of Linear Equations and Inequalities in Two Variables 11

2. 1. Graph linear equations by plotting ordered pairs. 2. Find intercepts. 3. Graph linear equations of the form Ax + By = . 4. Graph linear equations of the form y = b or x = a . 5. Use a linear equation to model data. Objectives 1 1 .2 Graphing Linear Equations in Two Variables

3. y = 2( ) – 1 Example Graph the linear equation y = 2 x – 1. Note that although this equation is not of the form Ax + By = C , it could be. Therefore, it is linear. To graph it, we will first find two points by letting x = 0 and then y = 0. Graph by Plotting Ordered Pairs If x = 0, then The graph of any linear equation in two variables is a straight line. y = – 1 So, we have the ordered pair (0,–1). = 2 x – 1 If y = 0, then 1 = 2 x So, we have the ordered pair ( ½ ,0). + 1 + 1 ½ = x

4. y = 2( 1 ) – 1 Example ( cont ) Graph the linear equation y = 2 x – 1. Now we will find a third point (just as a check) by letting x = 1. Graph by Plotting Ordered Pairs If x = 1, then y = 1 So, we have the ordered pair (1,1). When we graph, all three points, (0,–1), ( ½,0), and (1,1), should lie on the same straight line.

5. Finding Intercepts To find the x -intercept, let y = 0 in the given equation and solve for x . Then ( x , 0) is the x -intercept. To find the y -intercept, let x = 0 in the given equation and solve for y . Then (0, y ) is the y -intercept. Find Intercepts

6. + y = 2 Example Find the intercepts for the graph of x + y = 2. Then draw the graph. To find the y -intercept, let x = 0; to find the x -intercept, let y = 0. Graphing a Linear Equation Using Intercepts y = 2 The y -intercept is (0, 2). x + = 2 x = 2 The x -intercept is (2, 0). Plotting the intercepts gives the graph.

7. –6( ) + 2 y = 0 Example Graph the linear equation –6 x + 2 y = 0. First, find the intercepts. Graphing an Equation with x - and y -Intercepts (0, 0) 2 y = 0 The y -intercept is (0, 0). The x -intercept is (0, 0). Since the x and y intercepts are the same (the origin), choose a different value for x or y . y = 0 –6 x + 2( ) = 0 –6 x = 0 x = 0

8. –6( 1 ) + 2 y = 0 Example ( cont ) Graph the linear equation –6 x + 2 y = 0. Let x = 1. Graphing an Equation with x - and y -Intercepts (0, 0) 2 y = 6 A second point is (1, 3). y = 3 –6 + 2 y = 0 +6 +6

9. Line through the Origin The graph of a linear equation of the form Ax + By = where A and B are nonzero real numbers, passes through the origin (0,0). Graph Linear Equations of the Form Ax + By = 0

10. Note that this is the graph of a horizontal line with y -intercept (0,–2). Example Graph y = –2. Graphing a Horizontal Line The expanded version of this linear equation would be · x + y = –2. Here, the y -coordinate is unaffected by the value of the x -coordinate. Whatever x -value we choose, the y -value will be –2. Thus, we could plot the points (–1, –2), (2,–2), (4,–2), etc.

11. Example Graph x – 1 = 0. Graphing a Vertical Line Add 1 to each side of the equation. x = 1. The x -coordinate is unaffected by the value of the y -coordinate. Thus, we could plot the points (1, –3 ), (1, 0), (1, 2), etc. Note that this is the graph of a vertical line with no y- intercept.

12. Horizontal and Vertical Lines The graph of y = b , where b is a real number, is a horizontal line with y -intercept (0, b ) and no x -intercept (unless the horizontal line is the x -axis itself). The graph of x = a , where a is a real number, is a vertical line with x -intercept ( a , 0) and no y -intercept (unless the vertical line is the y -axis itself). Graph Linear Equations of the Form y = k or x = k

13. Example Bob has owned and managed Bob’s Bagels for the past 5 years and has kept track of his costs over that time. Based on his figures, Bob has determined that his total monthly costs can be modeled by C = 0.75 x + 2500, where x is the number of bagels that Bob sells that month. Use Bob’s cost equation to determine his costs if he sells 1000 bagels next month, 4000 bagels next month. Use a Linear Equation to Model Data C = 0.75(1000) + 2500 C = $3250 C = 0.75(4000) + 2500 C = $5500

14. Example 7 ( cont ) (b) Write the information from part (a) as two ordered pairs and use them to graph Bob’s cost equation. Use a Linear Equation to Model Data From part (a) we have (1000, 3250) and (4000, 5500). Note that we did not extend the graph to the left beyond the vertical axis. That area would correspond to a negative number of bagels, which does not make sense.