1. 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.

2. 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. y x 1 5 3 O –1
(1, 2)
SOLUTION
x + 3y = 6
Write original equation.
1 + 3(2) = 6 ?
Substitute 1 for x and 2 for y.
7  6
Simplify. Not a true statement.
(1, 2) is not a solution of the equation x + 3y = 6, so it is not on the graph.

3. 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. y x 1 5 3 O –1
(–3, 3)
SOLUTION
x + 3y = 6
Write original equation.
–3 + 3(3) = 6 ?
Substitute –3 for x and 3 for y.
6 = 6
Simplify. True statement.
(–3, 3) is a solution of the equation x + 3y = 6, so it is on the graph.

4. 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. y x 1 5 3 O –1
Then how can you ever graph an equation?
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.
Even then, the graph extends without limit to the left of the smallest input and to the right of the largest input.

5. 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. y x 1 5 3 O –1
Then how can you ever graph an equation?
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.
Even then, the graph extends without limit to the left of the smallest input and to the right of the largest input.
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.
This way you will see how the graph behaves to the left and right of the y -axis.

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

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

8. 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.
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
STEP 1
Rewrite the equation in function form, if necessary.
STEP 2
Choose a few values of x and make a table of values.
STEP 3
Plot the points from the table of values. A line through these points is the graph of the equation.

9. Graphing a Linear Equation
Use a table of values to graph the equation 3x + 2y = 1.
SOLUTION 1
Rewrite the equation in function form by solving for y. This will make it easier to make a table of values.
3x + 2y = 1
Write original equation.
2y = –3x + 1
Subtract 3x from each side.
y = – x + 3 2 1
Divide each side by 2. 2
Choose a few values for x and make a table of values.
Choose x.
Evaluate y. –2 2 7 2 1 2 5 2 –

10. ( ) ( ) Graphing a Linear Equation
Use a table of values to graph the equation 3x + 2y = 1.
SOLUTION
Evaluate y.
Choose x. –2 2 7 1 5 –
With the table of values you have found three solutions. 7 2 1 5 –
–2, , 0, , 2,
( )
( ) y x 1 2 3 O –2 –3 –4  3
Plot the points and draw a line through them. 
The graph of 3x + 2y = 1 is shown at the right. 

11. 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
Make a table of values.
Use 0 £ t £ 6, for 1996 – 2002. t h
14.90 1
21.66
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. 2
28.42 3
35.18 4
41.94 5
48.70 6
55.46

12. 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
y = b
x = a
In the coordinate plane, the graph y = b is a horizontal line.
In the coordinate plane, the graph x = a is a horizontal line.

13. Graph the following equation.
Graphing y = b
Graph the following equation.
y = 2
SOLUTION y x 1 2 3 O –2 –3 –4
The equation does not have x as a variable. 
(–3, 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:
y = 2
(–3, 2), (0, 2), (3, 2)
The graph of the equation is a horizontal line 2 units above the x-axis.

14. Graph the following equation.
Graphing x = a
Graph the following equation.
x = –3
SOLUTION y x 1 2 3 O –2 –3 –4
x = –3
The x-value is always –3, regardless of the value of y. 
(–3, 3)
For instance, here are some points that are solutions of the equation:
(–3, 0) 
(–3, –2) 
(–3, –2), (–3, 0), (–3, 3)
The graph of the equation is a vertical line 3 units to the left of the y-axis.