## 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 $x\text{- and }y\text{-values}$ 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.

$y=5x - 1$
$x$ $y$ $\left(x,y\right)$
$0$ $-1$ $\left(0,-1\right)$
$1$ $4$ $\left(1,4\right)$

To find a third solution, we’ll let $x=2$ and solve for $y$.

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

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

$y=5x - 1$
$x$ $y$ $\left(x,y\right)$
$0$ $-1$ $\left(0,-1\right)$
$1$ $4$ $\left(1,4\right)$
$2$ $9$ $\left(2,9\right)$

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 $y=4x - 2\text{:}$

$y=4x - 2$
$x$ $y$ $\left(x,y\right)$
$0$
$-1$
$2$

Solution
Substitute $x=0,x=-1$, and $x=2$ into $y=4x - 2$.

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

The results are summarized in the table.

$y=4x - 2$
$x$ $y$ $\left(x,y\right)$
$0$ $-2$ $\left(0,-2\right)$
$-1$ $-6$ $\left(-1,-6\right)$
$2$ $6$ $\left(2,6\right)$

### example

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

$5x - 4y=20$
$x$ $y$ $\left(x,y\right)$
$0$
$0$
$5$

### 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 $3x+2y=6$

### 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 $3x+2y=6$.

### example

Find three more solutions to the equation $3x+2y=6$

### try it

Let’s find some solutions to another equation now.

### example

Find three solutions to the equation $x - 4y=8$.

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.