1. Example 1: Finding Solutions of Equations with Two Variables
Use the given x-values to write solutions of the equation y = 4x + 2 as ordered pairs. x = 1, 2, 3, 4.
Make a function table by using the given values for x to find values for y.
Write these solutions as ordered pairs.
(x, y)‏ 18
4(4) + 2 4 14
4(3) + 2 3 10
4(2) + 2 2 6
4(1) + 2 1 y
4x + 2 x
(1, 6)‏
(2, 10)‏
(3, 14)‏
(4, 18)‏

2. Check It Out: Example 1
Use the given x-values to write solutions of the equation y = 3x + 2 as ordered pairs. x = 2, 3, 4, 5.
Make a function table by using the given values for x to find values for y.
Write these solutions as ordered pairs. 17
3(5) + 2 5 14
3(4) + 2 4 11
3(3) + 2 3 8
3(2) + 2 2 y
3x + 2 x
(x, y)‏
(2, 8)‏
(3, 11)‏
(4, 14)‏
(5, 17)‏

3. Check if an ordered pair is a solution of an equation by putting the x and y values into the equation to see if they make it a true statement.

4. Additional Example 2: Checking Solutions of Equations with Two Variables
Determine whether the ordered pair is a solution to the given equation.
(3, 21); y = 7x
Write the equation.
y = 7x
21 = 7(3)‏ ?
Substitute 3 for x and 21 for y.
21 = 21 ? 
So (3, 21) is a solution to y = 7x.

5. Check It Out: Example 2
Determine whether the ordered pair is a solution to the given equation.
(4, 20); y = 5x
Write the equation.
y = 5x
20 = 5(4)‏ ?
Substitute 4 for x and 20 for y.
20 = 20 ? 
So (4, 20) is a solution to y = 5x.

6. You can also graph the solutions of an equation on a coordinate plane
You can also graph the solutions of an equation on a coordinate plane. When you graph the ordered pairs of some functions, they form a straight line. The equations that express these functions are called linear equations.

7. Additional Example 3: Reading Solutions on Graphs
Use the graph of the linear function to find the value of y for the given value of x.
x = 4
Start at the origin and move 4 units right. x y 4
Move up until you reach the graph. Move left to find the y-value on the y-axis. 2 -2
When x = 4, y = 2.
The ordered pair is (4, 2). -4

8. Start at the origin and move 2 units right.
Check It Out: Example 3
Use the graph of the linear function to find the value of y for the given value of x.
x = 2
Start at the origin and move 2 units right. x y 4
Move up until you reach the graph. Move left to find the y-value on the y-axis. 2 -2
When x = 2, y = 4.
The ordered pair is (2, 4). -4

9. Additional Example 4: Graphing Linear Functions
Graph the function described by the equation. y = x + 2
Make a function table.
Write these solutions as ordered pairs. 3
(1) + 2 1 2
(0) + 2
(–1) + 2 –1 y
x + 2 x
(x, y)‏
(–1, 1)‏
(0, 2)‏
(1, 3)‏

10. Additional Example 4 Continued
Graph the ordered pairs on a coordinate plane. x y 5 4
Draw a line through the points to represent all the values of x you could have chosen and the corresponding values of y. 3 2 1 -1 -2 -3 -4 -5

11. Check It Out: Example 4
Graph the function described by the equation. y = 2x-1
Make a function table.
Write these solutions as ordered pairs. 1
2 (1) –1 –1
2 (0) –1 –5
2(-2) –1 –2 y
2x –1 x
(x, y)‏
(–2, –5)‏
(0, –1)‏
(1, 1)‏

12. Check It Out: Example 4 Continued
Graph the ordered pairs on a coordinate plane. x y 5 4
Draw a line through the points to represent all the values of x you could have chosen and the corresponding values of y. 3 2 1 -1 -2 -3 -4 -5

13. An equation in function form:
always starts with y =
If an equation is written another way, you can rewrite it in function form
Ex. 3x + y = 15
Subtract 3x from each side
-3x x
y = -3x + 15

14. Rewrite the equations in function form.
-6x + y = 14
-3x - y = 24
-6x + 2y = 14
8x - 2y = 16