3C Creating a Table of Ordered Pair Solutions to Linear Equations

Learning Outcomes

Complete a table of values that satisfy a linear equation
Find any solution to a linear 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.

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

try it

example

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

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

Show Solution

Solution

The figure shows three algebraic substitutions into an equation. The first substitution is x = 0, with 0 shown in blue. The next line is 5 x- 4 y = 20. The next line is 5 times 0, shown in blue - 4 y = 20. The next line is 0 - 4 y = 20. The next line is - 4 y = 20. The next line is y = -5. The last line is
The results are summarized in the table.

$5x - 4y=20$
$x$	$y$	$\left(x,y\right)$
$0$	$-5$	$\left(0,-5\right)$
$4$	$0$	$\left(4,0\right)$
$8$	$5$	$\left(8,5\right)$

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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 $3x+2y=6$

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Solution

Step 1: Choose any value for one of the variables in the equation.

We can substitute any value we want for

x

$x$ or any value for

y

$y$ .

Let’s pick $x=0$ .

What is the value of $y$ if $x=0$ ?

Step 2: Substitute that value into the equation.

Solve for the other variable.

Substitute

0

$0$ for

x

$x$ .

Simplify.

Divide both sides by $2$ .

3 x + 2 y = 6

$3x+2y=6$

$3\cdot\color{blue}{0}+2y=6$

$0+2y=6$

$2y=6$

$y=3$

Step 3: Write the solution as an ordered pair.

So, when

x = 0, y = 3

$x=0,y=3$ .

This solution is represented by the ordered pair

(0, 3)

$\left(0,3\right)$ .

Step 4: Check.

Substitute

x = 0, y = 3

$x=\color{blue}{0}, y=\color{red}{3}$ into the equation

3 x + 2 y = 6

$3x+2y=6$

Is the result a true equation?

Yes!

3 x + 2 y = 6

$3x+2y=6$

$3\cdot\color{blue}{0}+2\cdot\color{red}{3}\stackrel{?}{=}6$

$0+6\stackrel{?}{=}6$

$6=6\checkmark$

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 $3x+2y=6$

Show Solution

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

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

Check your answers.

(2, 0)

$\left(2,0\right)$

(1, \frac{3}{2})

$\left(1,\Large\frac{3}{2}\normalsize\right)$

(4, - 3)

$\left(4,-3\right)$

3 x + 2 y = 6

$3x+2y=6$

$3\cdot\color{blue}{2}+2\cdot\color{red}{0}\stackrel{?}{=}6$

$6+0\stackrel{?}{=}6$

$6+6\checkmark$

3 x + 2 y = 6

$3x+2y=6$

$3\cdot\color{blue}{1}+2\cdot\color{red}{\Large\frac{3}{2}}\normalsize\stackrel{?}{=}6$

$3+3\stackrel{?}{=}6$

$6+6\checkmark$

3 x + 2 y = 6

$3x+2y=6$

$3\cdot\color{blue}{4}+2\cdot\color{red}{-3}\stackrel{?}{=}6$

$12+(-6)\stackrel{?}{=}6$

$6+6\checkmark$

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

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

try it

Let’s find some solutions to another equation now.

example

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

Show Solution

Solution

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

So $\left(0,-2\right),\left(8,0\right)$ , and $\left(20,3\right)$ are three solutions to the equation $x - 4y=8$ .

$x - 4y=8$
$x$	$y$	$\left(x,y\right)$
$0$	$-2$	$\left(0,-2\right)$
$8$	$0$	$\left(8,0\right)$
$20$	$3$	$\left(20,3\right)$

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.

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Learning Outcomes

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example

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Find Solutions to Linear Equations in Two Variables

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