Graphing a Linear Equation A solution of an equation in two variables x and y is an ordered pair ( x, y ) that makes the equation true. The graph of an equation in x and y is the set of all points ( x, y ) that are solutions of the equation. In this lesson you will see that the graph of a linear equation is a line.

Verifying Solutions of an Equation Use the graph to decide whether the point lies on the graph of x + 3y = 6. Justify your answer algebraically. (1, 2) SOLUTION Write original equation. x + 3y = 6 Substitute 1 for x and 2 for y (2) = 6 ? Simplify. Not a true statement. 7  6 (1, 2) is not a solution of the equation x + 3y = 6, so it is not on the graph. y x O –1 1

Verifying Solutions of an Equation Use the graph to decide whether the point lies on the graph of x + 3y = 6. Justify your answer algebraically. (–3, 3) SOLUTION Write original equation. x + 3y = 6 –3 + 3(3) = 6 ? Simplify. True statement. 6 = 6 (–3, 3) is a solution of the equation x + 3y = 6, so it is on the graph. y x O –1 1 Substitute –3 for x and 3 for y.

One way is to make a table or a list of a few values, plot enough solutions to recognize a pattern, and then connect the points. Graphing a Linear Equation In the previous example you learned that the point (–3, 3) is on the graph of x + 3y = 6, but how many points does a graph have in all? The answer is that most graphs have too many points to list. Then how can you ever graph an equation? Even then, the graph extends without limit to the left of the smallest input and to the right of the largest input. y x O –1 1

One way is to make a table or a list of a few values, plot enough solutions to recognize a pattern, and then connect the points. When you make a table of values to graph an equation, you may want to choose values for x that include negative values, zero, and positive values. Graphing a Linear Equation In the previous example you learned that the point (–3, 3) is on the graph of x + 3y = 6, but how many points does a graph have in all? The answer is that most graphs have too many points to list. Then how can you ever graph an equation? Even then, the graph extends without limit to the left of the smallest input and to the right of the largest input. This way you will see how the graph behaves to the left and right of the y -axis. y x O –1 1

Choose x. Substitute to find the corresponding y -value. Graphing an Equation Use a table of values to graph the equation y + 2 = 3x. SOLUTION Write original equation. y + 2 = 3x Subtract 2 from each side. y = 3x – 2 Rewrite the equation in function form by solving for y. Choose a few values for x and make a table of values. y = 3(2) – 2 = 2 y = 3(1) – 2 = 1 y = 3(0) – 2 = 0 y = 3(–1) – 2 = –1 y = 3(–2) – 2 = –2 – 8– 8 –5 –2 1 4

Graphing an Equation Use a table of values to graph the equation y + 2 = 3x. SOLUTION Plot the points. Note that they appear to lie on a straight line. With the table of values you have found the five solutions (–2, –8 ), (–1, –5 ), ( 0, –2 ), ( 1, 1 ), and ( 2, 4 ). The line through the points is the graph of the equation. yx –1 –2 – 8 –5 –2 1 4      –1 y x 1 –33 1 –6 –8 3

Graphing a Linear Equation You may remember examples of linear equations in one variable. The solution of an equation such as 2 x – 1 = 3 is a real number. Its graph is a point on the real number line. GRAPHING A LINEAR EQUATION Rewrite the equation in function form, if necessary. Choose a few values of x and make a table of values. Plot the points from the table of values. A line through these points is the graph of the equation. STEP The equation y + 2 = 3x in the previous example is a linear equation in two variables. Its graph is a straight line.

Graphing a Linear Equation Use a table of values to graph the equation 3x + 2y = 1. SOLUTION Write original equation. 3x + 2y = 1 Subtract 3x from each side. 2y = –3x + 1 Rewrite the equation in function form by solving for y. This will make it easier to make a table of values. Choose a few values for x and make a table of values. 1 Choose x. Evaluate y. 2 Divide each side by 2. y = – x – –

Graphing a Linear Equation Use a table of values to graph the equation 3x + 2y = 1. SOLUTION Plot the points and draw a line through them. 3 With the table of values you have found three solutions. The graph of 3x + 2y = 1 is shown at the right. y x O –2 –3 – – –2,, 0,, 2, ( ) Evaluate y. Choose x. – –   

Using the Graph of a Linear Model An Internet Service Provider estimates that the number of households h (in millions) with Internet access can be modeled by h = 6.76 t , where t represents the number of years since Graph this model. Describe the graph in the context of the real-life situation. SOLUTION th From the table and the graph, you can see that the number of households with Internet access is projected to increase by about 7 million households per year. Make a table of values. Use 0  t  6, for 1996 – 2002.

Horizontal and Vertical Lines All linear equations in x and y can be written in the form A x + B y = C. When A = 0 the equation reduces to B y = C and the graph is a horizontal line. When B = 0 the equation reduces to A x = C and the graph is a vertical line. EQUATIONS OF HORIZONTAL AND VERTICAL LINES In the coordinate plane, the graph y = b is a horizontal line. In the coordinate plane, the graph x = a is a horizontal line. y = b x = a

y = 2 SOLUTION (–3, 2), (0, 2), (3, 2) The graph of the equation is a horizontal line 2 units above the x -axis. The equation does not have x as a variable. Graphing y = b Graph the following equation. y x O –2 –3 –4  (–3, 2) y = 2 (0, 2)  (3, 2)  The y -value is always 2, regardless of the value of x. For instance, here are some points that are solutions of the equation:

x = –3 SOLUTION (–3, –2), (–3, 0), (–3, 3) The graph of the equation is a vertical line 3 units to the left of the y -axis. The x -value is always –3, regardless of the value of y. Graphing x = a Graph the following equation. y x O –2 –3 –4  (–3, 3) x = –3 (–3, 0)  (–3, –2)  For instance, here are some points that are solutions of the equation: