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]

Show Solution

[latex]5x - 4y=20[/latex]
[latex]0[/latex]	[latex]-5[/latex]	[latex]\left(0,-5\right)[/latex]
[latex]4[/latex]	[latex]0[/latex]	[latex]\left(4,0\right)[/latex]
[latex]8[/latex]	[latex]5[/latex]	[latex]\left(8,5\right)[/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]

Show Solution

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]

Show Solution

Solution
To find solutions to [latex]3x+2y=6[/latex], choose a value for [latex]x[/latex] or [latex]y[/latex]. Remember, we can choose any value we want for [latex]x[/latex] or [latex]y[/latex]. Here we chose [latex]1[/latex] for [latex]x[/latex], and [latex]0[/latex] and [latex]-3[/latex] for [latex]y[/latex].

Substitute it into the equation.	[latex]y=\color{red}{0}[/latex] [latex]3x+2y=6[/latex] [latex]3x+2(\color{red}{0})=6[/latex]	[latex]y=\color{blue}{1}[/latex] [latex]3x+2y=6[/latex] [latex]3(\color{blue}{1})+2y=6[/latex]	[latex]y=\color{red}{-3}[/latex] [latex]3x+2y=6[/latex] [latex]3x+2(\color{red}{-3})=6[/latex]
Simplify. Solve.	[latex]3x+0=6[/latex] [latex]3x=6[/latex]	[latex]3+2y=6[/latex] [latex]2y=3[/latex]	[latex]3x-6=6[/latex] [latex]3x=12[/latex]
	[latex]x=2[/latex]	[latex]y=\Large\frac{3}{2}[/latex]	[latex]x=4[/latex]
Write the ordered pair.	[latex]\left(2,0\right)[/latex]	[latex]\left(1,\Large\frac{3}{2}\normalsize\right)[/latex]	[latex]\left(4,-3\right)[/latex]

Check your answers.

[latex]\left(2,0\right)[/latex]

[latex]\left(1,\Large\frac{3}{2}\normalsize\right)[/latex]

[latex]\left(4,-3\right)[/latex]

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

[latex]3\cdot\color{blue}{2}+2\cdot\color{red}{0}\stackrel{?}{=}6[/latex]

[latex]6+0\stackrel{?}{=}6[/latex]

[latex]6+6\checkmark[/latex]

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

[latex]3\cdot\color{blue}{1}+2\cdot\color{red}{\Large\frac{3}{2}}\normalsize\stackrel{?}{=}6[/latex]

[latex]3+3\stackrel{?}{=}6[/latex]

[latex]6+6\checkmark[/latex]

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

[latex]3\cdot\color{blue}{4}+2\cdot\color{red}{-3}\stackrel{?}{=}6[/latex]

[latex]12+(-6)\stackrel{?}{=}6[/latex]

[latex]6+6\checkmark[/latex]

So [latex]\left(2,0\right),\left(1,\Large\frac{3}{2}\normalsize\right)[/latex] and [latex]\left(4,-3\right)[/latex] are all solutions to the equation [latex]3x+2y=6[/latex]. In the previous example, we found that [latex]\left(0,3\right)[/latex] is a solution, too. We can list these solutions in a table.

[latex]3x+2y=6[/latex]
[latex]x[/latex]	[latex]y[/latex]	[latex]\left(x,y\right)[/latex]
[latex]0[/latex]	[latex]3[/latex]	[latex]\left(0,3\right)[/latex]
[latex]2[/latex]	[latex]0[/latex]	[latex]\left(2,0\right)[/latex]
[latex]1[/latex]	[latex]\Large\frac{3}{2}[/latex]	[latex]\left(1,\Large\frac{3}{2}\normalsize\right)[/latex]
[latex]4[/latex]	[latex]-3[/latex]	[latex]\left(4,-3\right)[/latex]

try it

Let’s find some solutions to another equation now.

example

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

Show Solution

[latex]x - 4y=8[/latex]
[latex]0[/latex]	[latex]-2[/latex]	[latex]\left(0,-2\right)[/latex]
[latex]8[/latex]	[latex]0[/latex]	[latex]\left(8,0\right)[/latex]
[latex]20[/latex]	[latex]3[/latex]	[latex]\left(20,3\right)[/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.

[latex]x-4y=8[/latex]	[latex]x-4y=8[/latex]	[latex]x-4y=8[/latex]
Choose a value for [latex]x[/latex] or [latex]y[/latex].	[latex]x=\color{blue}{0}[/latex]	[latex]y=\color{red}{0}[/latex]	[latex]y=\color{red}{3}[/latex]
Substitute it into the equation.	[latex]\color{blue}{0}-4y=8[/latex]	[latex]x-4\cdot\color{red}{0}=8[/latex]	[latex]x-4\cdot\color{red}{3}=8[/latex]
Solve.	[latex]-4y=8[/latex] [latex]y=-2[/latex]	[latex]x-0=8[/latex] [latex]x=8[/latex]	[latex]x-12=8[/latex] [latex]x=20[/latex]
Write the ordered pair.	[latex]\left(0,-2\right)[/latex]	[latex]\left(8,0\right)[/latex]	[latex]\left(20,3\right)[/latex]

Chapter 5: Graphs