Copyright © Cengage Learning. All rights reserved. 3 Graphs; Equations of Lines; Functions; Variation Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. Graphs of Linear Equations in Two Variables Section 3.2 Copyright © Cengage Learning. All rights reserved.

Objectives Determine whether an ordered pair satisfies an equation in two variables. Construct a table of values given an equation. Graph a linear equation in two variables by constructing a table of values. Graph a linear equation in two variables using the intercept method. 1 2 3 4

Objectives Graph a horizontal line and a vertical line given an equation. Write a linear equation in two variables from given information, graph the equation, and interpret the graphed data. 5 6

Determine whether an ordered pair satisfies an equation in two variables 1.

Equations in 2 Variables The equation x + 2y = 5 contains the two variables x and y solutions of such equations are ordered pairs of numbers (x = 1, y = 2) For example, the ordered pair (1, 2) is a solution, because the equation is satisfied when x = 1 and y = 2. x + 2y = 5 1 + 2(2) = 5 1 + 4 = 5 5 = 5

Example Is the pair (–2, 4) a solution of y = 3x + 9? Solution: We substitute –2 for x and 4 for y and determine whether the resulting equation is true. y = 3x + 9 4 ≟ 3(–2) + 9 4 ≟ –6 + 9

Example – Solution cont’d 4 = 3 Since the equation 4 = 3 is false, the pair (–2, 4) is not a solution.

Construct a table of values given an equation 2.

Table of Solutions to an Equation To find solutions of equations in x and y, we can pick numbers at random, substitute them for x, and find the corresponding values of y. For example, to find some ordered pairs that satisfy y = 5 – x, we can let x = 1(called the input value), substitute 1 for x, and solve for y (called the output value). y = 5 – x

Table of Solutions to an Equation y = 5 – 1 y = 4 The ordered pair (1, 4) is a solution. As we find solutions, we will list them in a table of values like Table (1) below.

Table of Solutions to an Equation If x = 2, we have y = 5 – x y = 5 – 2 y = 3 A second solution is (2, 3). We list it in Table (2) below. This is the original equation. Substitute the input value of 2 for x. The output is 3.

Table of Solutions to an Equation If x = 5, we have y = 5 – x y = 5 – 5 y = 0 A third solution is (5, 0). We list it in Table (3) below. This is the original equation. Substitute the input value of 5 for x. The output is 0.

Table of Solutions to an Equation If x = –1, we have y = 5 – x y = 5 – (–1) y = 6 A fourth solution is (–1, 6). We list it in Table (4) below. This is the original equation. Substitute the input value of –1 for x. The output is 6.

Table of Solutions to an Equation If x = 6, we have y = 5 – x y = 5 – 6 y = –1 A fifth solution is (6, –1). We list it in Table (5) below. This is the original equation. Substitute the input value of 6 for x. The output is –1.

Table of Solutions to an Equation Since we can choose any real number for x, and since any choice of x will give a corresponding value of y, we can see that the equation y = 5 – x has infinitely many solutions.

Graph a linear equation in two variables by constructing a table of values 3.

Graphing a Linear Equation in 2 Variables A linear equation is any equation that can be written in the form Ax + By = C, where A, B, and C are real numbers and A and B are not both 0. To graph the equation y = 5 – x, we plot the ordered pairs listed in the table on a rectangular coordinate system, as in Figure 3-10. Figure 3-10

Graphing a Linear Equation in 2 Variables From the figure, we can see that the five points lie on a line. We draw a line through the points. The arrowheads on the line show that the graph continues forever in both directions. Since the graph of any solution of y = 5 – x will lie on this line, the line is a picture of all of the solutions of the equation. The line is said to be the graph of the equation. Any equation, such as y = 5 – x, whose graph is a line is called a linear equation in two variables. Any point on the line has coordinates that satisfy the equation, and the graph of any pair (x, y) that satisfies the equation is a point on the line.

Graphing a Linear Equation in 2 Variables Since we usually will choose a number for x first and then find the corresponding value of y, the value of y depends on x. For this reason, we call y the dependent variable and x the independent variable. The value of the independent variable is the input value, and the value of the dependent variable is the output value. Although only two points are needed to graph a linear equation, we often plot a third point as a check. If the three points do not lie on a line, at least one of them is in error.

Graphing a Linear Equation in 2 Variables Graphing Linear Equations in Two Variables Find two ordered pairs (x, y) that satisfy the equation by choosing arbitrary input values for x and solving for the corresponding output values of y. A third point provides a check. Plot each resulting pair (x, y) on a rectangular coordinate system. If they do not lie on a line, check your calculations. Draw the line passing through the points.

Example Graph by constructing a table of values and plotting points: y = 3x – 4. Solution: We find three ordered pairs that satisfy the equation. If x = 1 If x = 2 If x = 3 y = 3x – 4 y = 3x – 4 y = 3x – 4 y = 3(1) – 4 y = 3(2) – 4 y = 3(3) – 4 y = 1 y = 2 y = 5

Example – Solution cont’d We enter the results in a table of values, plot the points, and draw a line through the points. The graph appears in Figure 3-11. Figure 3-11

Graph a linear equation in two variables using the intercept method 4.

Graphing Linear Equation Using the Intercept Method The points where a line intersects the x- and y-axes are called intercepts of the line. x- and y-Intercepts The x-intercept of a line is a point (a, 0) where the line intersects the x-axis. (See Figure 3-14.) To find a, substitute 0 for y in the equation of the line and solve for x. Figure 3-14

Graphing Linear Equation Using the Intercept Method A y-intercept of a line is a point (0, b) where the line intersects the y-axis. To find b, substitute 0 for x in the equation of the line and solve for y. Plotting the x- and y-intercepts and drawing a line through them is called the intercept method of graphing a line. This method is useful for graphing equations written in general form.

Graphing Linear Equation Using the Intercept Method General Form of the Equation of a Line If A, B, and C are real numbers and A and B are not both 0, then the equation Ax + By = C is called the general form of the equation of a line. Comment Whenever possible, we will write the general form Ax + By = C so that A, B, and C are integers and A  0. We also will make A, B, and C as small as possible. For example, the equation 6x + 12y = 24 can be written as x + 2y = 4 by dividing both sides by 6.

Example Graph by using the intercept method: 3x + 2y = 6. Solution: To find the y-intercept, we let x = 0 and solve for y. 3x + 2y = 6 3(0) + 2y = 6 2y = 6 y = 3 The y-intercept is the point with coordinates (0, 3). Substitute 0 for x. Simplify. Divide both sides by 2.

Example – Solution cont’d To find the x-intercept, we let y = 0 and solve for x. 3x + 2y = 6 3x + 2(0) = 6 3x = 6 x = 2 The x-intercept is the point with coordinates (2, 0). As a check, we plot one more point. If x = 4, then 3(4) + 2y = 6 12 + 2y = 6 2y = –6 Substitute 0 for y. Simplify. Divide both sides by 3. Substitute 4 for x. Simplify. Subtract 12 from both sides.

Example – Solution cont’d y = –3 The point (4, –3) is on the graph. We plot these three points and join them with a line. The graph of 3x + 2y = 6 is shown in Figure 3-15. Figure 3-15

Graph a horizontal line and a vertical line given an equation 5.

Horizontal and Vertical Lines Equations such as y = 3 and x = –2 are linear equations, because they can be written in the general form Ax + By = C. y = 3 is equivalent to 0x + 1y = 3 x = –2 is equivalent to 1x + 0y = –2

Example Graph: a. y = 3 b. x = –2. Solution: a. We can write the equation y = 3 in general form as 0x + y = 3. Since the coefficient of x is 0, the numbers chosen for x have no effect on y. The value of y is always 3. For example, if we substitute –3 for x, we get 0x + y = 3 0(–3) + y = 3

Example – Solution cont’d 0 + y = 3 y = 3 The table in Figure 3-16(a) gives several pairs that satisfy the equation y = 3. Figure 3-16(a)

Example – Solution cont’d After plotting these pairs and joining them with a line, we see that the graph of y = 3 is a horizontal line that intersects the y-axis at 3. The y-intercept is (0, 3). There is no x-intercept. b. We can write x = –2 in general form as x + 0y = –2. Since the coefficient of y is 0, the values of y have no effect on x. The value of x is always –2.

Example – Solution cont’d A table of values and the graph are shown in Figure 3-16(b). The graph of x = –2 is a vertical line that intersects the x-axis at –2. The x-intercept is (–2, 0). There is no y-intercept. Figure 3-16(b)

Graphing Linear Equation Using the Intercept Method Equations of Horizontal and Vertical Lines Suppose a and b are real numbers. The equation y = b represents a horizontal line that intersects the y-axis at (0, b). If b = 0, the line is the x-axis. The equation x = a represents a vertical line that intersects the x-axis at (a, 0). If a = 0, the line is the y-axis.

Write a linear equation in two variables from given information, graph the equation, and interpret the graphed data 6.

Linear Equation in 2 Variables We have solved applications using one variable. In the next example, we will write an equation containing two variables to describe an application and then graph the equation.

Example – Birthday Parties A restaurant offers a party package that includes food, drinks, cake, and party favors for a cost of $25 plus $3 per child. Write a linear equation that will give the cost for a party of any size. Graph the equation and determine the meaning of the y-intercept in the context of this problem. Solution: Cost of party: c Number of children: n c = 25 + 3n

Example – Solution cont’d c = 25 + 3n

Example – Solution If n = 0 If n = 5 If n = 10 cont’d If n = 0 If n = 5 If n = 10 c = 25 + 3(0) c = 25 + 3(5) c = 25 + 3(10) c = 25 c = 25 + 15 c = 25 + 30 c = 40 c = 55 The results are recorded in the table.

Example – Solution cont’d Next, we graph the points in Figure 3-17 and draw a line through them. We don’t draw an arrowhead on the left, because it doesn’t make sense to have a negative number of children attend a party. From the graph, we can determine the y-intercept is (0, 25). The $25 represents the setup cost for a party with no attendees. Figure 3-17

Example – Solution Comment cont’d Comment The scale for the cost (y-axis) is 5 units and the scale for the number attending (x-axis) is 1. Since the scales on the x- and y-axes are not the same, you must label them!