Learning Outcomes
- Complete a table of values that satisfy a linear equation
- Find any solution to a linear equation
In the previous examples, we substituted the x- and y-values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for x and then solve the equation for y. Or, choose a value for y and then solve for x.
We’ll start by looking at the solutions to the equation y=5x−1 we found in the previous chapter. We can summarize this information in a table of solutions.
y=5x−1 |
x |
y |
(x,y) |
0 |
−1 |
(0,−1) |
1 |
4 |
(1,4) |
|
|
|
To find a third solution, we’ll let x=2 and solve for y.
|
y=5x−1 |
Substitute x=2 |
y=5(2)−1 |
Multiply. |
y=10−1 |
Simplify. |
y=9 |
The ordered pair is a solution to y=5x−1. We will add it to the table.
y=5x−1 |
x |
y |
(x,y) |
0 |
−1 |
(0,−1) |
1 |
4 |
(1,4) |
2 |
9 |
(2,9) |
We can find more solutions to the equation by substituting any value of x or any value of y and solving the resulting equation to get another ordered pair that is a solution. There are an infinite number of solutions for this equation.
example
Complete the table to find three solutions to the equation y=4x−2:
y=4x−2 |
x |
y |
(x,y) |
0 |
|
|
−1 |
|
|
2 |
|
|
Solution
Substitute x=0,x=−1, and x=2 into y=4x−2.
x=0 |
x=−1 |
x=2 |
y=4x−2 |
y=4x−2 |
y=4x−2 |
y=4⋅0−2 |
y=4(−1)−2 |
y=4⋅2−2 |
y=0−2 |
y=−4−2 |
y=8−2 |
y=−2 |
y=−6 |
y=6 |
(0,−2) |
(−1,−6) |
(2,6) |
The results are summarized in the table.
y=4x−2 |
x |
y |
(x,y) |
0 |
−2 |
(0,−2) |
−1 |
−6 |
(−1,−6) |
2 |
6 |
(2,6) |
example
Complete the table to find three solutions to the equation 5x−4y=20:
5x−4y=20 |
x |
y |
(x,y) |
0 |
|
|
|
0 |
|
|
5 |
|
Show Solution
Solution

The results are summarized in the table.
5x−4y=20 |
x |
y |
(x,y) |
0 |
−5 |
(0,−5) |
4 |
0 |
(4,0) |
8 |
5 |
(8,5) |
Find Solutions to Linear Equations in Two Variables
To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either x or y. We could choose 1,100,1,000, or any other value we want. But it’s a good idea to choose a number that’s easy to work with. We’ll usually choose 0 as one of our values.
example
Find a solution to the equation 3x+2y=6
Show Solution
Solution
Step 1: Choose any value for one of the variables in the equation. |
We can substitute any value we want for x or any value for y.
Let’s pick x=0.
What is the value of y if x=0 ? |
Step 2: Substitute that value into the equation.
Solve for the other variable. |
Substitute 0 for x.
Simplify.
Divide both sides by 2. |
3x+2y=6
3⋅0+2y=6
0+2y=6
2y=6
y=3 |
Step 3: Write the solution as an ordered pair. |
So, when x=0,y=3. |
This solution is represented by the ordered pair (0,3). |
Step 4: Check. |
Substitute x=0,y=3 into the equation 3x+2y=6
Is the result a true equation?
Yes! |
3x+2y=6
3⋅0+2⋅3?=6
0+6?=6
6=6✓ |
We said that linear equations in two variables have infinitely many solutions, and we’ve just found one of them. Let’s find some other solutions to the equation 3x+2y=6.
example
Find three more solutions to the equation 3x+2y=6
Show Solution
Solution
To find solutions to 3x+2y=6, choose a value for x or y. Remember, we can choose any value we want for x or y. Here we chose 1 for x, and 0 and −3 for y.
Substitute it into the equation. |
y=0
3x+2y=6
3x+2(0)=6 |
y=1
3x+2y=6
3(1)+2y=6 |
y=−3
3x+2y=6
3x+2(−3)=6 |
Simplify.
Solve. |
3x+0=6
3x=6 |
3+2y=6
2y=3 |
3x−6=6
3x=12 |
|
x=2 |
y=32 |
x=4 |
Write the ordered pair. |
(2,0) |
(1,32) |
(4,−3) |
Check your answers.
(2,0) |
(1,32) |
(4,−3) |
3x+2y=6
3⋅2+2⋅0?=6
6+0?=6
6+6✓ |
3x+2y=6
3⋅1+2⋅32?=6
3+3?=6
6+6✓ |
3x+2y=6
3⋅4+2⋅−3?=6
12+(−6)?=6
6+6✓ |
So (2,0),(1,32) and (4,−3) are all solutions to the equation 3x+2y=6. In the previous example, we found that (0,3) is a solution, too. We can list these solutions in a table.
3x+2y=6 |
x |
y |
(x,y) |
0 |
3 |
(0,3) |
2 |
0 |
(2,0) |
1 |
32 |
(1,32) |
4 |
−3 |
(4,−3) |
Let’s find some solutions to another equation now.
example
Find three solutions to the equation x−4y=8.
Show Solution
Solution
x−4y=8 |
x−4y=8 |
x−4y=8 |
Choose a value for x or y. |
x=0 |
y=0 |
y=3 |
Substitute it into the equation. |
0−4y=8 |
x−4⋅0=8 |
x−4⋅3=8 |
Solve. |
−4y=8
y=−2 |
x−0=8
x=8 |
x−12=8
x=20 |
Write the ordered pair. |
(0,−2) |
(8,0) |
(20,3) |
So (0,−2),(8,0), and (20,3) are three solutions to the equation x−4y=8.
x−4y=8 |
x |
y |
(x,y) |
0 |
−2 |
(0,−2) |
8 |
0 |
(8,0) |
20 |
3 |
(20,3) |
Remember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true.