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

	$y=5x - 1$
	$y=5(\color{blue}{2})-1$
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

[ohm_question]146945[/ohm_question] [ohm_question]146947[/ohm_question]

example

Complete the table to find three solutions to the equation

5x - 4y=20\text{:}

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

Answer: 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.

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

try it

[ohm_question]146948[/ohm_question]

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

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

Answer: Solution

Step 1: Choose any value for one of the variables in the equation.	We can substitute any value we want for $x$ or any value for $y$ . 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 $0$ for $x$ . Simplify. Divide both sides by $2$ .	$3x+2y=6$ $3\cdot\color{blue}{0}+2y=6$ $0+2y=6$ $2y=6$ $y=3$
Step 3: Write the solution as an ordered pair.	So, when $x=0,y=3$ .	This solution is represented by the ordered pair $\left(0,3\right)$ .
Step 4: Check.	Substitute $x=\color{blue}{0}, y=\color{red}{3}$ into the equation $3x+2y=6$ Is the result a true equation? Yes!	$3x+2y=6$ $3\cdot\color{blue}{0}+2\cdot\color{red}{3}\stackrel{?}{=}6$ $0+6\stackrel{?}{=}6$ $6=6\checkmark$

try it

[ohm_question]147000[/ohm_question]

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

Answer: 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.	$3x+0=6$ $3x=6$	$3+2y=6$ $2y=3$	$3x-6=6$ $3x=12$
$x=2$	$y=\frac{3}{2}$	$x=4$
Write the ordered pair.	$\left(2,0\right)$	$\left(1,\frac{3}{2}\right)$	$\left(4,-3\right)$

Check your answers.

$\left(2,0\right)$	$\left(1,\frac{3}{2}\right)$	$\left(4,-3\right)$
$3x+2y=6$ $3\cdot\color{blue}{2}+2\cdot\color{red}{0}\stackrel{?}{=}6$ $6+0\stackrel{?}{=}6$ $6+6\checkmark$	$3x+2y=6$ $3\cdot\color{blue}{1}+2\cdot\color{red}{\frac{3}{2}}\stackrel{?}{=}6$ $3+3\stackrel{?}{=}6$ $6+6\checkmark$	$3x+2y=6$ $3\cdot\color{blue}{4}+2\cdot\color{red}{-3}\stackrel{?}{=}6$ $12+(-60\stackrel{?}{=}6$ $6+6\checkmark$

\left(2,0\right),\left(1,\frac{3}{2}\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.

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

try it

[ohm_question]147003[/ohm_question]

Let’s find some solutions to another equation now.

example

Find three solutions to the equation

x - 4y=8

Answer: 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.	$\left(0,-2\right)$	$\left(8,0\right)$	$\left(20,3\right)$

\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.

TRY IT

[ohm_question]147004[/ohm_question]

Creating a Table of Ordered Pair Solutions to a Linear Equation

Learning Outcomes

example

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example

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Find Solutions to Linear Equations in Two Variables

example

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example

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example

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Studienführer > Prealgebra

Creating a Table of Ordered Pair Solutions to a Linear Equation

Learning Outcomes

example

try it

example

try it

Find Solutions to Linear Equations in Two Variables

example

try it

example

try it

example

TRY IT

Licenses & Attributions

CC licensed content, Original

CC licensed content, Specific attribution