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] |
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
Solution
The results are summarized in the table.
| [latex]5x - 4y=20[/latex] |
| [latex]x[/latex] |
[latex]y[/latex] |
[latex]\left(x,y\right)[/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] |
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
Solution
| Step 1: Choose any value for one of the variables in the equation. |
We can substitute any value we want for [latex]x[/latex] or any value for [latex]y[/latex].
Let’s pick [latex]x=0[/latex].
What is the value of [latex]y[/latex] if [latex]x=0[/latex] ? |
| Step 2: Substitute that value into the equation.
Solve for the other variable. |
Substitute [latex]0[/latex] for [latex]x[/latex].
Simplify.
Divide both sides by [latex]2[/latex]. |
[latex]3x+2y=6[/latex]
[latex]3\cdot\color{blue}{0}+2y=6[/latex]
[latex]0+2y=6[/latex]
[latex]2y=6[/latex]
[latex]y=3[/latex] |
| Step 3: Write the solution as an ordered pair. |
So, when [latex]x=0,y=3[/latex]. |
This solution is represented by the ordered pair [latex]\left(0,3\right)[/latex]. |
| Step 4: Check. |
Substitute [latex]x=\color{blue}{0}, y=\color{red}{3}[/latex] into the equation [latex]3x+2y=6[/latex]
Is the result a true equation?
Yes! |
[latex]3x+2y=6[/latex]
[latex]3\cdot\color{blue}{0}+2\cdot\color{red}{3}\stackrel{?}{=}6[/latex]
[latex]0+6\stackrel{?}{=}6[/latex]
[latex]6=6\checkmark[/latex] |
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] |
Let’s find some solutions to another equation now.
example
Find three solutions to the equation [latex]x - 4y=8[/latex].
Show Solution
Solution
| [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] |
So [latex]\left(0,-2\right),\left(8,0\right)[/latex], and [latex]\left(20,3\right)[/latex] are three solutions to the equation [latex]x - 4y=8[/latex].
| [latex]x - 4y=8[/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]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.
