Copyright © 2008 Pearson Education, Inc Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Graph of Linear Equations 11 Graph of Linear Equations 11.1 Graphs and Applications of Linear Equations 11.2 More with Graphing and Intercepts Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

GRAPHS and APPLICATIONS OF LINEAR EQUATIONS 11.1 GRAPHS and APPLICATIONS OF LINEAR EQUATIONS a Plot points associated with ordered pairs of numbers, determine the quadrant in which a point lies. Find the coordinates of a point on a graph. Determine whether an ordered pair is a solution of an equation with two variables. Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. Solve applied problems involving graphs of linear equations. b c d e Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Objective a Plot points associated with ordered pairs of numbers, determine the quadrant in which a point lies. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Points and Ordered Pairs To graph, or plot, points we use two perpendicular number lines called axes. The point at which the axes cross is called the origin. Arrows on the axes indicate the positive directions. Consider the pair (2, 3). The numbers in such a pair are called the coordinates. The first coordinate in this case is 2 and the second coordinate is 3. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Points and Ordered Pairs continued To plot the point (2, 3) we start at the origin. Move 2 units in the horizontal direction. The second number 3, is positive. We move 3 units in the vertical direction (up). Make a “dot” and label the point. (2, 3) (2, 3) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example A Plot the point (4, 3). Solution Starting at the origin, we move 4 units in the negative horizontal direction. The second number, 3, is positive, so we move 3 units in the positive vertical direction (up). (4, 3) 3 units up 4 units left Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The horizontal and vertical axes divide the plane into four regions, or quadrants. In which quadrant is the point (3, 4) located? IV In which quadrant is the point (3, 4) located? III Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Objective Find the coordinates of a point on a graph. b Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example B Find the coordinates of points A, B, C, D, E, F, and G. Solution Point A is 5 units to the right of the origin and 3 units above the origin. Its coordinates are (5, 3). The other coordinates are as follows: B: (2, 4) C: (3, 4) D: (3, 2) E: (2, 3) F: (3, 0) G: (0, 2) A B C D E F G Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Objective c Determine whether an ordered pair is a solution of an equation with two variables. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example C Determine whether each of the following pairs is a solution of 4y + 3x = 18: a) (2, 3); b)(1, 5). Solution a) We substitute 2 for x and 3 for y. 4y + 3x = 18 4•3 + 3•2 | 18 12 + 6 | 18 18 = 18 True b) We substitute 1 for x and 5 for y. 4•5 + 3•1 | 18 20 + 3 | 18 23 = 18 False Since 18 = 18 is true, the pair (2, 3) is a solution. Since 23 = 18 is false, the pair (1, 5) is not a solution. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Objective d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Graph of An Equation The graph of an equation is a drawing that represents all its solutions. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Graph a Linear Equation 1. Select a value for one variable and calculate the corresponding value of the other value. Form an ordered pair using alphabetical order as indicated by the variables. 2. Repeat step (1) to obtain at least two other ordered pairs. Two points are essential to determine a straight line. A third ordered point serves as a check. 3. Plot the ordered pairs and draw a straight line passing through the points. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example D Graph y = 3x Solution Find some ordered pairs that are solutions. We choose any number for x and then determine y by substitution. y x y = 3x (x, y) 2 1 1 2 (2, 6) 6 (2, 6) 3 (1, 3) (1, 3) (0, 0) (0, 0) 3 (1, 3) 6 (2, 6) 1. Choose x. 2. Compute y. 4. Plot the points. 3. Form the ordered pair (x, y). Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example E Graph y = 4x + 1 Solution We select convenient values for x and compute y, and form an ordered pair. If x = 2, then y = 4(2) + 1 = 7 and (2, 7) is a solution. If x = 0, then y = 4(0) + 1 = 1 and (0, 1) is a solution. If x = 2, then y = 4(2) + 1 = 9 and (2, 9) is a solution. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution (continued) Results are often listed in a table. (1) Choose x. (2) Compute y. (3) Form the pair (x, y). (4) Plot the points. x y (x, y) 2 7 (2, 7) 1 (0, 1) 2 9 (2, 9) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution (continued) Note that all three points line up. If they didn’t we would know that we had made a mistake. Finally, use a ruler or other straightedge to draw a line. Every point on the line represents a solution of y = 4x + 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

y-Intercept The graph of the equation y = mx + b passes through the y-intercept (0, b). (0, b) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example F Graph Solution Complete a table of values. y (x, y) 4 (4, 4) 3 (0, 3) 4 2 (4, 2) y-intercept (4, 4) (4, 2) We see that (0, 3) is a solution. It is the y-intercept. (0, 3) is the y-intercept. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example G Graph and identify the y-intercept. Solution To find an equivalent equation in the form y = mx + b, we solve for y: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

continued Complete a table of values. x y (x, y) (0, 0) 2 3 (2, 3) (0, 0) 2 3 (2, 3) 2 3 (2, 3)  y-intercept y-intercept (2, 3) (2, 3) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Objective Solve applied problems involving graphs of linear equations. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example H The cost c, in dollars, of shipping a FedEx Priority Overnight package weighing 1 lb or more a distance of 1001 to 1400 mi is given by c = 2.8w + 21.05 where w is the package’s weight in pounds. Graph the equation and then use the graph to estimate the cost of shipping a 10 ½-pound package. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution Select values for w and then calculate c. c = 2.8w + 21.05 If w = 2, then c = 2.8(2) + 21.05 = 26.65 If w = 4, then c = 2.8(4) + 21.05 = 32.25 If w = 8, then c = 2.8(8) + 21.05 = 43.45 w c 2 26.65 4 32.25 8 43.45 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution (continued) Plot the points. To estimate an 10 ½ pound package, we locate the point on the line that is above 10 ½ and then find the value on the c-axis that corresponds to that point. The cost of shipping an 10 ½ pound package is about $51.00. Weight (in pounds) Mail cost (in dollars) 10 ½ pounds Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

More with Graphing and Intercepts 11.2 More with Graphing and Intercepts a Find the intercepts of a linear equation, and graph using intercepts. Graph equations equivalent to those of the type x = a and y = b. b Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Objective a Find the intercepts of a linear equation, and graph using intercepts. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Intercepts The y-intercept is (0, b). To find b, let x = 0 and solve the original equation for y. The x-intercept is (a, 0). To find a, let y = 0 and solve the original equation for x. (0, b) (a, 0) y-intercept x-intercept Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example A Consider 5x + 2y = 10. Find the intercepts Example A Consider 5x + 2y = 10. Find the intercepts. Then graph the equation using the intercepts. Solution To find the y-intercept, we let x = 0 and solve for y: 5 • 0 + 2y = 10 2y = 10 y = 5 The y-intercept is (0, 5). To find the x-intercept, we let y = 0 and solve for x. 5x + 2• 0 = 10 5x = 10 x = 2 The x-intercept is (2, 0). Replacing x with 0 Replacing y with 0 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

continued We plot these points and draw the line, or graph. A third point should be used as a check. We substitute any convenient value for x and solve for y. If we let x = 4, then 5 • 4 + 2y = 10 20 + 2y = 10 2y = 10 y = 5 5x + 2y = 10 x-intercept (2, 0) y-intercept (0, 5) x y 5 2 4 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example B Graph y = 4x Solution We know that (0, 0) is both the x-intercept and y-intercept. We calculate two other points and complete the graph, knowing it passes through the origin. y = 4x (1, 4) (1, 4) (0, 0) x y 1 4 1 4 x-intercept y-intercept Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Objective b Graph equations equivalent to those of the type x = a and y = b. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example C Graph y = 2 y = 2 Solution We regard the equation y = 2 as 0 • x + y = 2. No matter what number we choose for x, we find that y must equal 2. y = 2 Choose any number for x. y must be 2. x y (x, y) 2 (0, 2) 4 (4, 2) 4 (4 , 2) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

continued y = 2 Solution When we plot the ordered pairs (0, 2), (4, 2) and (4, 2) and connect the points, we obtain a horizontal line. Any ordered pair of the form (x, 2) is a solution, so the line is parallel to the x-axis with y-intercept (0, 2). y = 2 (4, 2) (0, 2) (4, 2) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example D Graph x = 2 Solution We regard the equation x = 2 as x + 0 • y = 2. We make up a table with all 2 in the x-column. x = 2 x must be 2. x y (x, y) 2 4 (2, 4) 1 (2, 1) 4 (2, 4) Any number can be used for y. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

continued x = 2 Solution When we plot the ordered pairs (2, 4), (2, 1), and (2, 4) and connect them, we obtain a vertical line. Any ordered pair of the form (2, y) is a solution. The line is parallel to the y-axis with x-intercept (2, 0). x = 2 (2, 4) (2, 4) (2, 1) Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Horizontal and Vertical Lines The graph of y = b is a horizontal line. The y-intercept is (0, b). The graph of x = a is a vertical line. The x-intercept is (a, 0). Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley