All the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one solution. The process of solving an equation ended with a statement such as [latex]x=4[/latex]. Then we checked the solution by substituting back into the equation.
Here’s an example of a linear equation in one variable, and its one solution.
[latex-display]\begin{array}{c}3x+5=17\hfill \\ \\ 3x=12\hfill \\ x=4\hfill \end{array}[/latex-display]
But equations can have more than one variable. Equations with two variables can be written in the general form [latex]Ax+By=C[/latex]. An equation of this form is called a linear equation in two variables.
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-display]x+4y=8[/latex-display]
[latex-display]\color{blue}{0}+4\cdot\color{red}{2}\stackrel{?}{=}8[/latex-display]
[latex-display]0+8\stackrel{?}{=}8[/latex-display]
[latex]8=8\checkmark[/latex] |
[latex]x=\color{blue}{2}, y=\color{red}{-4}[/latex]
[latex-display]x+4y=8[/latex-display]
[latex-display]\color{blue}{2}+4(\color{red}{-4})\stackrel{?}{=}8[/latex-display]
[latex-display]2+(-16)\stackrel{?}{=}8[/latex-display]
[latex]-14=8[/latex] |
[latex]x=\color{blue}{-4}, y=\color{red}{3}[/latex]
[latex-display]x+4y=8[/latex-display]
[latex-display]\color{blue}{-4}+4\cdot\color{red}{3}\stackrel{?}{=}8[/latex-display]
[latex-display]-4+12\stackrel{?}{=}8[/latex-display]
[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. |
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]
Answer:
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-display]y=5x-1[/latex-display]
[latex-display]\color{red}{-1}\stackrel{?}{=}5(\color{blue}{o})-1[/latex-display]
[latex-display]-1\stackrel{?}{=}0-1[/latex-display]
[latex]-1=-1\checkmark[/latex] |
[latex]x=\color{blue}{1}, y=\color{red}{4}[/latex]
[latex-display]y=5x-1[/latex-display]
[latex-display]\color{red}{4}\stackrel{?}{=}5(\color{blue}{1})-1[/latex-display]
[latex-display]4\stackrel{?}{=}5-1[/latex-display]
[latex]4=4\checkmark[/latex] |
[latex]x=\color{blue}{-2}, y=\color{red}{-7}[/latex]
[latex-display]y=5x-1[/latex-display]
[latex-display]\color{red}{-7}\stackrel{?}{=}5(\color{blue}{-2})-1[/latex-display]
[latex-display]-7\stackrel{?}{=}-10-1[/latex-display]
[latex]-7=-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. |