Solving Equations By Clearing Decimals

Learning Outcomes

Determine the LCD of an equation that contains decimals
Solve equations with decimals that require several steps

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But decimals are really another way to represent fractions. For example, [latex]0.3=\Large\frac{3}{10}[/latex] and [latex]0.17=\Large\frac{17}{100}[/latex]. So, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the least common denominator.

Example

Solve: [latex]0.8x - 5=7[/latex]

Solution:
The only decimal in the equation is [latex]0.8[/latex]. Since [latex]0.8=\Large\frac{8}{10}[/latex], the LCD is [latex]10[/latex]. We can multiply both sides by [latex]10[/latex] to clear the decimal.

	[latex]0.8x-5=7[/latex]
Multiply both sides by the LCD.	[latex]\color{red}{10}(0.8x-5)=\color{red}{10}(7)[/latex]
Distribute.	[latex]10(0.8x)-10(5)=10(7)[/latex]
Multiply, and notice, no more decimals!	[latex]8x-50=70[/latex]
Add 50 to get all constants to the right.	[latex]8x-50\color{red}{+50}=70\color{red}{+50}[/latex]
Simplify.	[latex]8x=120[/latex]
Divide both sides by [latex]8[/latex].	[latex]\Large\frac{8x}{\color{red}{8}}\normalsize =\Large\frac{120}{\color{red}{8}}[/latex]
Simplify.	[latex]x=15[/latex]
Check: Let [latex]x=15[/latex].
[latex]0.8(\color{red}{15})-5\stackrel{\text{?}}{=}7[/latex] [latex]12-5\stackrel{\text{?}}{=}7[/latex] [latex]7=7\quad\checkmark[/latex]

Try it

Example

Solve: [latex]0.06x+0.02=0.25x - 1.5[/latex]

Show Solution

Try it

In the following video we present another example of how to solve an equation that contains decimals and variable terms on both sides of the equal sign.

The next example uses an equation that is typical of the ones we will see in the money applications. Notice that we will distribute the decimal first before we clear all decimals in the equation.

Example

Solve: [latex]0.25x+0.05\left(x+3\right)=2.85[/latex]

Show Solution

	[latex]0.06x+0.02=0.25x-1.5[/latex]
Multiply both sides by 100.	[latex]\color{red}{100}(0.06x+0.02)=\color{red}{100}(0.25x-1.5)[/latex]
Distribute.	[latex]100(0.06x)+100(0.02)=100(0.25x)-100(1.5)[/latex]
Multiply, and now no more decimals.	[latex]6x+2=25x-150[/latex]
Collect the variables to the right.	[latex]6x\color{red}{-6x}+2=25x\color{red}{-6x}-150[/latex]
Simplify.	[latex]2=19x-150[/latex]
Collect the constants to the left.	[latex]2\color{red}{+150}=19x-150\color{red}{+150}[/latex]
Simplify.	[latex]152=19x[/latex]
Divide by [latex]19[/latex].	[latex]\Large\frac{152}{\color{red}{19}}\normalsize =\Large\frac{19x}{\color{red}{19}}[/latex]
Simplify.	[latex]8=x[/latex]
Check: Let [latex]x=8[/latex].
[latex]0.06(\color{red}{8})+0.02=0.25(\color{red}{8})-1.5[/latex] [latex]0.48+0.02=2.00-1.5[/latex] [latex]0.50=0.50\quad\checkmark[/latex]

	[latex]0.25x+0.05(x+3)=2.85[/latex]
Distribute first.	[latex]0.25x+0.05x+0.15=2.85[/latex]
Combine like terms.	[latex]0.30x+0.15=2.85[/latex]
To clear decimals, multiply by [latex]100[/latex].	[latex]\color{red}{100}(0.30x+0.15)=\color{red}{100}(2.85)[/latex]
Distribute.	[latex]30x+15=285[/latex]
Subtract [latex]15[/latex] from both sides.	[latex]30x+15\color{red}{-15}=285\color{red}{-15}[/latex]
Simplify.	[latex]30x=270[/latex]
Divide by [latex]30[/latex].	[latex]\Large\frac{30x}{\color{red}{30}}\normalsize =\Large\frac{270}{\color{red}{30}}[/latex]
Simplify.	[latex]x=9[/latex]
Check: Let [latex]x=9[/latex].
[latex]0.25x+0.05(x+3)=2.85[/latex] [latex]0.25(\color{red}{9})+0.05(\color{red}{9}+3)\stackrel{\text{?}}{=}2.85[/latex] [latex]2.25+0.05(12)\stackrel{\text{?}}{=}2.85[/latex] [latex]2.85=2.85\quad\checkmark[/latex]

Module 5: Multi-Step Linear Equations