Learning Objectives
- Determine whether a given ordered pair is a solution of a given linear equation.
- Find solutions of a linear equation.
- Complete a table of solutions.
Key words
- Ordered pair solution: a solution written in the form [latex]\left (x,y\right )[/latex]
Finding Solutions of Linear Equations in Two Variables
When an equation has two variables, any solution will be an ordered pair with a value for each variable.
Solution to a Linear Equation in Two Variables
An ordered pair [latex]\left(x,y\right)[/latex] is a solution of the linear equation [latex]ax+by=c[/latex], if the equation is a true statement when the [latex]x[/latex]– and [latex]y[/latex]-values of the ordered pair are substituted into the equation.
Example
Determine whether [latex](−2,4)[/latex] is a solution of the equation [latex]4y+5x=3[/latex].
Solution
Substitute [latex]x=−2[/latex] and [latex]y=4[/latex] into the equation:
[latex]\begin{array}{r}4y+5x=3\\4\left(4\right)+5\left(−2\right)=3\end{array}[/latex]
Evaluate.
[latex]\begin{array}{r}16+\left(−10\right)=3\\6=3\end{array}[/latex]
The statement is not true, so [latex](−2,4)[/latex] is not a solution.
Answer
[latex](−2,4)[/latex] is not a solution of the equation [latex]4y+5x=3[/latex].
example
Determine which ordered pairs are solutions of the equation [latex]x+4y=8\text{:}[/latex]
1. [latex]\left(0,2\right)[/latex]
2. [latex]\left(2,-4\right)[/latex]
3. [latex]\left(-4,3\right)[/latex]
Solution
Substitute the [latex]x\text{- and }y\text{-values}[/latex] from each ordered pair into the equation and determine if the result is a true statement.
| 1. [latex]\left(0,2\right)[/latex] | 2. [latex]\left(2,-4\right)[/latex] | 3. [latex]\left(-4,3\right)[/latex] |
| [latex]x=\color{blue}{0}, y=\color{red}{2}[/latex][latex]x+4y=8[/latex]
[latex]\color{blue}{0}+4\cdot\color{red}{2}\stackrel{?}{=}8[/latex] [latex]0+8\stackrel{?}{=}8[/latex] [latex]8=8\checkmark[/latex] |
[latex]x=\color{blue}{2}, y=\color{red}{-4}[/latex][latex]x+4y=8[/latex]
[latex]\color{blue}{2}+4(\color{red}{-4})\stackrel{?}{=}8[/latex] [latex]2+(-16)\stackrel{?}{=}8[/latex] [latex]-14\not=8[/latex] |
[latex]x=\color{blue}{-4}, y=\color{red}{3}[/latex][latex]x+4y=8[/latex]
[latex]\color{blue}{-4}+4\cdot\color{red}{3}\stackrel{?}{=}8[/latex] [latex]-4+12\stackrel{?}{=}8[/latex] [latex]8=8\checkmark[/latex] |
| [latex]\left(0,2\right)[/latex] is a solution. | [latex]\left(2,-4\right)[/latex] is not a solution. | [latex]\left(-4,3\right)[/latex] is a solution. |
try it
example
Determine which ordered pairs are solutions of the equation. [latex]y=5x - 1\text{:}[/latex]
1. [latex]\left(0,-1\right)[/latex]
2. [latex]\left(1,4\right)[/latex]
3. [latex]\left(-2,-7\right)[/latex]
Solution
Substitute the [latex]x\text{-}[/latex] and [latex]y\text{-values}[/latex] from each ordered pair into the equation and determine if it results in a true statement.
| 1. [latex]\left(0,-1\right)[/latex] | 2. [latex]\left(1,4\right)[/latex] | 3. [latex]\left(-2,-7\right)[/latex] | [latex]x=\color{blue}{0}, y=\color{red}{-1}[/latex][latex]y=5x-1[/latex]
[latex]\color{red}{-1}\stackrel{?}{=}5(\color{blue}{0})-1[/latex] [latex]-1\stackrel{?}{=}0-1[/latex] [latex]-1=-1\checkmark[/latex] |
[latex]x=\color{blue}{1}, y=\color{red}{4}[/latex][latex]y=5x-1[/latex]
[latex]\color{red}{4}\stackrel{?}{=}5(\color{blue}{1})-1[/latex] [latex]4\stackrel{?}{=}5-1[/latex] [latex]4=4\checkmark[/latex] |
[latex]x=\color{blue}{-2}, y=\color{red}{-7}[/latex][latex]y=5x-1[/latex]
[latex]\color{red}{-7}\stackrel{?}{=}5(\color{blue}{-2})-1[/latex] [latex]-7\stackrel{?}{=}-10-1[/latex] [latex]-7\not=-11[/latex] |
| [latex]\left(0,-1\right)[/latex] is a solution. | [latex]\left(1,4\right)[/latex] is a solution. | [latex]\left(-2,-7\right)[/latex] is not a solution. |
try it
The video shows more examples of how to determine whether an ordered pair is a solution of a linear equation.
Complete a Table of Solutions
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 of a given 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].
Let’s consider the equation [latex]y=5x - 1[/latex]. The easiest value to choose for [latex]x[/latex] or [latex]y[/latex] is zero:
[latex]\begin{equation}\begin{aligned}y & =5x-1 \;\;\;\;\;\;\;\;\;\;\text{Substitute}\;x=0\\y & = 5(0)-1\\y & = -1\end{aligned}\end{equation}[/latex] So, [latex]x=0,\;y=-1[/latex] is a solution, which as an ordered pair is [latex]\left (0,\,-1\right )[/latex].
[latex]\begin{equation}\begin{aligned}y & =5x-1 \;\;\;\;\;\;\;\;\;\text{Substitute}\;y=0\\0 & = 5x-1\;\;\;\;\;\;\;\;\;\text{Solve for}\;x\\1 & = 5x\\ \frac{1}{5} & =x\end{aligned}\end{equation}[/latex] So, [latex]x=\frac{1}{5},\;y=0[/latex] is a solution, which as an ordered pair is [latex]\left (\frac{1}{5},\,0\right )[/latex].
We can continue to find more solutions by choosing different values of [latex]x[/latex] and [latex]y[/latex].
Suppose [latex]x=2[/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] |
To find a third solution, we’ll let [latex]x=2[/latex] and solve for [latex]y[/latex].
We can write our solutions in a 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]\frac{1}{5}[/latex] | [latex]0[/latex] | [latex]\left(\frac{1}{5},0\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 of 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] | ||
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] |
try it
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, -\frac{4}{5}, 2.6[/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]
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] |
try it
try it
example
Find three solutions to the equation [latex]x - 4y=8[/latex].
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 ordered pair we find is a solution if it makes the equation true.
TRY IT
