3.3 Solve Equations with Variables and Constants on Both Sides

In this equation, the variable is found only on the left side. It makes sense to call the left side the “variable” side. Therefore, the right side will be the “constant” side. We will write the labels above the equation to help us remember what goes where.

[latex]\overset{{\color{red}{v}}{\color{red}{a}}{\color{red}{r}}{\color{red}{i}}{\color{red}{a}}{\color{red}{b}}{\color{red}{l}}{\color{red}{e}}}{7x+8}=\overset{{\color{red}{c}}{\color{red}{o}}{\color{red}{n}}{\color{red}{s}}{\color{red}{t}}{\color{red}{a}}{\color{red}{n}}{\color{red}{t}}}{-13}[/latex]

Since the left side is the “[latex]x[/latex]”, or variable side, the [latex]8[/latex] is out of place. We must “undo” adding 8 by subtracting 8, and to keep the equality we must subtract [latex]8[/latex] from both sides.

Step 1: Use the Subtraction Property of Equality.

[latex]\begin{align*}&\;&7x\;+\;8\;{\color{red}{-}}{\color{red}{\;}}{\color{red}{8}}\;&=\;-13\;{\color{red}{-}}{\color{red}{\;}}{\color{red}{8}}\\&\text{Simplify}\;&7x\;&=\;-21&\end{align*}[/latex]

Now all the variables are on the left and the constant on the right.
The equation looks like those you learned to solve earlier.

Step 2: Use the Division Property of Equality.

[latex]\begin{align*}&\;&\frac{7x}{\color{red}{7}}&=\frac{-21}{\color{red}{7}}\\ &\text{Simplify}\;&x&=-3\end{align*}[/latex]

Step 3: Check:

[latex]7x+8=-13[/latex]

Step 4: Let [latex]x=-3[/latex].

[latex]\begin{align*}7\left({\color{red}{-}}{\color{red}{3}}\right)+8&\overset?=-13\\ -21+8&\overset?=-13 \\ -13 &= -13\checkmark \end{align*}[/latex]

Here the variable is on both sides, but the constants only appear on the right side, so let’s make the right side the “constant” side. Then the left side will be the “variable” side.

[latex]\overset{{\color{red}{v}}{\color{red}{a}}{\color{red}{r}}{\color{red}{i}}{\color{red}{a}}{\color{red}{b}}{\color{red}{l}}{\color{red}{e}}}{9x}=\overset{{\color{red}{c}}{\color{red}{o}}{\color{red}{n}}{\color{red}{s}}{\color{red}{t}}{\color{red}{a}}{\color{red}{n}}{\color{red}{t}}}{8x-6}[/latex]

Step 1: We don’t want any [latex]x[/latex]’s on the right, so subtract the [latex]8x[/latex] from both sides.

[latex]\begin{align*}&\;&9x{\color{red}{-}}{\color{red}{8}}{\color{red}{x}}&=8x{\color{red}{-}}{\color{red}{8}}{\color{red}{x}}-6\\ &\text{Simplify}\;&x&=6\end{align*}[/latex]

We succeeded in getting the variables on one side and the constants on the other, and have obtained the solution.

Step 3: Check:

[latex]9x=8x-6[/latex]

Step 4: Let [latex]x=-6[/latex].

[latex]\begin{align*}9x\left({\color{red}{-}}{\color{red}{6}}\right)&\overset?=8x\left({\color{red}{-}}{\color{red}{6}}\right)-6 \\ -54&\overset?=-48-6 \\ -54 &= -54\checkmark \end{align*}[/latex]

The only constant is on the left and the [latex]y[/latex]’s are on both sides. Let’s leave the constant on the left and get the variables to the right.

[latex]\overset{{\color{red}{c}}{\color{red}{o}}{\color{red}{n}}{\color{red}{s}}{\color{red}{t}}{\color{red}{a}}{\color{red}{n}}{\color{red}{t}}}{5y-9}=\overset{{\color{red}{v}}{\color{red}{a}}{\color{red}{r}}{\color{red}{i}}{\color{red}{a}}{\color{red}{b}}{\color{red}{l}}{\color{red}{e}}}{8y}[/latex]

Step 1: Subtract [latex]5y[/latex] from both sides.

[latex]\begin{align*}&\;&5y{\color{red}{-}}{\color{red}{5}}{\color{red}{y}}-9&=8y{\color{red}{-}}{\color{red}{5}}{\color{red}{y}}\\ &\text{Simplify}\;&-9&=3y\end{align*}[/latex]

Step 2: We have the [latex]y[/latex]’s on the right and the constants on the left. Divide both sides by [latex]3[/latex].

[latex]\begin{align*}&\;&\frac{-9}{\color{red}{3}}&=\frac{3y}{\color{red}{3}}\\ &\text{Simplify}\;&-3&=y\end{align*}[/latex]

Step 3: Check:

[latex]5y-9=8y[/latex]

Step 4: Let [latex]y=-3[/latex].

[latex]\begin{align*} 5\left({\color{red}{-}}{\color{red}{3}}\right)-9&\overset?=8\left({\color{red}{-}}{\color{red}{3}}\right) \\ -15-9&\overset?=-24 \\ -24 &= -24\checkmark \end{align*}[/latex]

Since [latex]\frac{5}{4}>\frac{1}{4}[/latex], make the left side the “variable” side and the right side the “constant” side.

[latex]\overset{{\color{red}{v}}{\color{red}{a}}{\color{red}{r}}{\color{red}{i}}{\color{red}{a}}{\color{red}{b}}{\color{red}{l}}{\color{red}{e}}}{\frac54x+6}=\overset{{\color{red}{c}}{\color{red}{o}}{\color{red}{n}}{\color{red}{s}}{\color{red}{t}}{\color{red}{a}}{\color{red}{n}}{\color{red}{t}}}{\frac14x-2}[/latex]

Step 1: Subtract [latex]\frac{1}{4}x[/latex] from both sides.

[latex]\begin{align*}&\;&\frac54x{\color{red}{-}}{\color{red}{\frac14}}{\color{red}{x}}+6&=\frac14x{\color{red}{-}}{\color{red}{\frac14}}{\color{red}{x}}-2\\ &\text{Combine like terms.}\;&x+6&=-2 \end{align*}[/latex]

Step 2: Subtract [latex]6[/latex] from both sides.

[latex]{x+6}{\color{red}{-}}{{\color{red}{6}}=-2{\color{red}{-}}}{\color{red}{6}}{}[/latex]

Step 3: Simplify.

[latex]x=-8[/latex]

Step 4: Check: Let [latex]x=-8[/latex]

[latex]\begin{align*} \frac54x+6&=\frac14x-2\\ \frac54\left(-8\right)+6&\overset?=\frac14\left(-8\right)-2\\ -10+6&\overset?=-2-2\\ -4&=-4\checkmark \end{align*}[/latex]

Since [latex]7.8>5.4[/latex], make the left side the “variable” side and the right side the “constant” side.

[latex]\overset{{\color{red}{v}}{\color{red}{a}}{\color{red}{r}}{\color{red}{i}}{\color{red}{a}}{\color{red}{b}}{\color{red}{l}}{\color{red}{e}}{\color{red}{\;}}{\color{red}{s}}{\color{red}{i}}{\color{red}{d}}{\color{red}{e}}}{7.8x+4}=\overset{{\color{red}{c}}{\color{red}{o}}{\color{red}{n}}{\color{red}{s}}{\color{red}{t}}{\color{red}{a}}{\color{red}{n}}{\color{red}{t}}{\color{red}{\;}}{\color{red}{s}}{\color{red}{i}}{\color{red}{d}}{\color{red}{e}}}{5.4x-8}[/latex]

Step 1: Subtract [latex]5.4x[/latex] from both sides.

[latex]\begin{align*}&\;&7.8x{\color{red}{-}}{\color{red}{5}}{\color{red}{.}}{\color{red}{4}}{\color{red}{x}}+4&=5.4x{\color{red}{-}}{\color{red}{5}}{\color{red}{.}}{\color{red}{4}}{\color{red}{x}}-8\\ &\text{Combine like terms.}\;&2.4x+4&=-8 \end{align*}[/latex]

Step 2: Subtract [latex]4[/latex] from both sides.

[latex]\begin{align*}&\;&2.4x+4{\color{red}{-}}{\color{red}{4}}&=-8{\color{red}{-}}{\color{red}{4}}\\ &\text{Simplify.}\;&2.4x&=-12 \end{align*}[/latex]

Step 3: Use the Division Property of Equality.

[latex]\begin{align*}&\;&\frac{2.4x}{{\color{red}{2}}{\color{red}{.}}{\color{red}{4}}}&=\frac{-12}{{\color{red}{2}}{\color{red}{.}}{\color{red}{4}}}\\ &\text{Simplify.}\;&x&=-5 \end{align*}[/latex]

Step 4: Check:

[latex]7.8x+4=5.4x-5[/latex]

Step 5: Let [latex]x=-5[/latex].

[latex]\begin{align*} 7.8{\color{red}{\left(-5\right)}}+4&=5.4{\color{red}{\left(-5\right)}}-8\\ -39+4&\overset?=-27-8\\ -35&=-35\checkmark \end{align*}[/latex]