Creating a Table of Ordered Pair Solutions to a Linear Equation

Learning Outcomes

  • Complete a table of values that satisfy a two variable equation
  • Find any solution to a two variable equation

In the previous examples, we substituted the [latex]x\text{- and }y\text{-values}[/latex] 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 [latex]x[/latex] and then solve the equation for [latex]y[/latex]. Or, choose a value for [latex]y[/latex] and then solve for [latex]x[/latex].

We’ll start by looking at the solutions to the equation [latex]y=5x - 1[/latex] we found in the previous chapter. We can summarize this information in a table of solutions.

[latex]y=5x - 1[/latex]
[latex]x[/latex] [latex]y[/latex] [latex]\left(x,y\right)[/latex]
[latex]0[/latex] [latex]-1[/latex] [latex]\left(0,-1\right)[/latex]
[latex]1[/latex] [latex]4[/latex] [latex]\left(1,4\right)[/latex]

To find a third solution, we’ll let [latex]x=2[/latex] and solve for [latex]y[/latex].

[latex]y=5x - 1[/latex]
Substitute [latex]x=2[/latex] [latex]y=5(\color{blue}{2})-1[/latex]
Multiply. [latex]y=10 - 1[/latex]
Simplify. [latex]y=9[/latex]

The ordered pair is a solution to [latex]y=5x - 1[/latex]. We will add it to the table.

[latex]y=5x - 1[/latex]
[latex]x[/latex] [latex]y[/latex] [latex]\left(x,y\right)[/latex]
[latex]0[/latex] [latex]-1[/latex] [latex]\left(0,-1\right)[/latex]
[latex]1[/latex] [latex]4[/latex] [latex]\left(1,4\right)[/latex]
[latex]2[/latex] [latex]9[/latex] [latex]\left(2,9\right)[/latex]

We can find more solutions to the equation by substituting any value of [latex]x[/latex] or any value of [latex]y[/latex] 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 [latex]y=4x - 2\text{:}[/latex]

[latex]y=4x - 2[/latex]
[latex]x[/latex] [latex]y[/latex] [latex]\left(x,y\right)[/latex]
[latex]0[/latex]
[latex]-1[/latex]
[latex]2[/latex]

Solution
Substitute [latex]x=0,x=-1[/latex], and [latex]x=2[/latex] into [latex]y=4x - 2[/latex].

[latex]x=\color{blue}{0}[/latex] [latex]x=\color{blue}{-1}[/latex] [latex]x=\color{blue}{2}[/latex]
[latex]y=4x - 2[/latex] [latex]y=4x - 2[/latex] [latex]y=4x - 2[/latex]
[latex]y=4\cdot{\color{blue}{0}}-2[/latex] [latex]y=4(\color{blue}{-1})-2[/latex] [latex]y=4\cdot{\color{blue}{2}}-2[/latex]
[latex]y=0 - 2[/latex] [latex]y=-4 - 2[/latex] [latex]y=8 - 2[/latex]
[latex]y=-2[/latex] [latex]y=-6[/latex] [latex]y=6[/latex]
[latex]\left(0,-2\right)[/latex] [latex]\left(-1,-6\right)[/latex] [latex]\left(2,6\right)[/latex]

The results are summarized in the table.

[latex]y=4x - 2[/latex]
[latex]x[/latex] [latex]y[/latex] [latex]\left(x,y\right)[/latex]
[latex]0[/latex] [latex]-2[/latex] [latex]\left(0,-2\right)[/latex]
[latex]-1[/latex] [latex]-6[/latex] [latex]\left(-1,-6\right)[/latex]
[latex]2[/latex] [latex]6[/latex] [latex]\left(2,6\right)[/latex]

 

try it

 

example

Complete the table to find three solutions to the equation [latex]5x - 4y=20\text{:}[/latex]

[latex]5x - 4y=20[/latex]
[latex]x[/latex] [latex]y[/latex] [latex]\left(x,y\right)[/latex]
[latex]0[/latex]
[latex]0[/latex]
[latex]5[/latex]

 

try it

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 [latex]x[/latex] or [latex]y[/latex]. We could choose [latex]1,100,1,000[/latex], 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 [latex]0[/latex] as one of our values.

example

Find a solution to the equation [latex]3x+2y=6[/latex]

 

try it

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 [latex]3x+2y=6[/latex].

example

Find three more solutions to the equation [latex]3x+2y=6[/latex]

 

try it

Let’s find some solutions to another equation now.

example

Find three solutions to the equation [latex]x - 4y=8[/latex].

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.

TRY IT