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Study Guides > Prealgebra

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 [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]
. [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]
   

try it

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

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]

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.

[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

[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 [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].

Answer: 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-display]3\cdot\color{blue}{0}+2y=6[/latex-display] [latex-display]0+2y=6[/latex-display] [latex-display]2y=6[/latex-display] [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-display]3\cdot\color{blue}{0}+2\cdot\color{red}{3}\stackrel{?}{=}6[/latex-display] [latex-display]0+6\stackrel{?}{=}6[/latex-display] [latex]6=6\checkmark[/latex]

 

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 [latex]3x+2y=6[/latex].  

example

Find three more solutions to the equation [latex]3x+2y=6[/latex].

Answer: 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-display]3x+2y=6[/latex-display] [latex]3x+2(\color{red}{0})=6[/latex] [latex]y=\color{blue}{1}[/latex] [latex-display]3x+2y=6[/latex-display] [latex]3(\color{blue}{1})+2y=6[/latex] [latex]y=\color{red}{-3}[/latex] [latex-display]3x+2y=6[/latex-display] [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=\frac{3}{2}[/latex] [latex]x=4[/latex]
Write the ordered pair. [latex]\left(2,0\right)[/latex] [latex]\left(1,\frac{3}{2}\right)[/latex] [latex]\left(4,-3\right)[/latex]
  Check your answers.
[latex]\left(2,0\right)[/latex] [latex]\left(1,\frac{3}{2}\right)[/latex] [latex]\left(4,-3\right)[/latex]
[latex]3x+2y=6[/latex] [latex-display]3\cdot\color{blue}{2}+2\cdot\color{red}{0}\stackrel{?}{=}6[/latex-display] [latex-display]6+0\stackrel{?}{=}6[/latex-display] [latex]6+6\checkmark[/latex] [latex]3x+2y=6[/latex] [latex-display]3\cdot\color{blue}{1}+2\cdot\color{red}{\frac{3}{2}}\stackrel{?}{=}6[/latex-display] [latex-display]3+3\stackrel{?}{=}6[/latex-display] [latex]6+6\checkmark[/latex] [latex]3x+2y=6[/latex] [latex-display]3\cdot\color{blue}{4}+2\cdot\color{red}{-3}\stackrel{?}{=}6[/latex-display] [latex-display]12+(-60\stackrel{?}{=}6[/latex-display] [latex]6+6\checkmark[/latex]
So [latex]\left(2,0\right),\left(1,\frac{3}{2}\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]\frac{3}{2}[/latex] [latex]\left(1,\frac{3}{2}\right)[/latex]
[latex]4[/latex] [latex]-3[/latex] [latex]\left(4,-3\right)[/latex]

 

try it

[ohm_question]147003[/ohm_question]
  Let’s find some solutions to another equation now.

example

Find three solutions to the equation [latex]x - 4y=8[/latex].

Answer: 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.

TRY IT

[ohm_question]147004[/ohm_question]
 

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  • Question ID 147004, 147003, 147000. Authored by: Lumen Learning. License: CC BY: Attribution.

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