Solving Equations with Fraction and Decimal Coefficients

Learning Outcomes

Solve equations with fraction coefficients
Solve equations with decimal coefficients

In this section, we will explore how to solve equations with fractions in them. To start, we will review how equations with fractions can be solved by multiplying by the reciprocal. We will then move on to learn how to use the Multiplication Property of Equality to eliminate the fractions from an equation, making it simpler to solve.

Solving equations with fractions by multiplying by the reciprocal

In our first example, we will solve a one-step equation using the multiplication property of equality. You will see that the variable is part of a fraction in the given equation, and using the multiplication property of equality allows us to remove the variable from the fraction. Remember that fractions imply division, so you can think of this as the variable [latex]k[/latex] is being divided by 10. To “undo” the division, you can use multiplication to isolate [latex]k[/latex]. (Note that there is a negative term in the equation, so it will be important to think about the sign of each term as you work through the problem. Stop after each step you take to make sure all the terms have the correct sign.)

Example

Solve [latex]-\frac{7}{2}=\frac{k}{10}[/latex] for [latex]k[/latex].

Show Solution

In our next example, we are given an equation that contains a variable multiplied by a fraction. We will use a reciprocal to isolate the variable.

example

Solve: [latex]\Large\frac{2}{3}\normalsize x=18[/latex]

Show Solution

Notice that we could have divided both sides of the equation [latex]\Large\frac{2}{3}\normalsize x=18[/latex] by [latex]\Large\frac{2}{3}[/latex] to isolate [latex]x[/latex]. While this would work, multiplying by the reciprocal requires fewer steps.

try it

In the following video, you will see examples of using the multiplication property of equality to solve a one-step equation involving negative fractions.

Solving equations with fractions by clearing the denominators

You may feel overwhelmed when you see fractions in an equation, so we are going to show a method to solve equations with fractions where you use the common denominator to eliminate the fractions from an equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions.

Pay attention to the fact that each term in the equation gets multiplied by the least common denominator. That’s what makes it equal to the original!

EXAMPLE

Solve: [latex]\Large\frac{1}{8}\normalsize x+\Large\frac{1}{2}=\Large\frac{1}{4}[/latex]

Solution:

	[latex]\Large\frac{1}{8}\normalsize x+\Large\frac{1}{2}=\Large\frac{1}{4}\normalsize\quad{LCD=8}[/latex]
Multiply both sides of the equation by that LCD, [latex]8[/latex]. This clears the fractions.	[latex]\color{red}{8(}\Large\frac{1}{8}\normalsize x+\Large\frac{1}{2}\color{red}{)}=\normalsize\color{red}{8(}\Large\frac{1}{4}\color{red}{)}[/latex]
Use the Distributive Property.	[latex]8\cdot\Large\frac{1}{8}\normalsize x+8\cdot\Large\frac{1}{2}\normalsize=8\cdot\Large\frac{1}{4}[/latex]
Simplify — and notice, no more fractions!	[latex]x+4=2[/latex]
Solve using the General Strategy for Solving Linear Equations.	[latex]x+4\color{red}{-4}=2\color{red}{-4}[/latex]
Simplify.	[latex]x=-2[/latex]
Check: Let [latex]x=-2[/latex][latex]\Large\frac{1}{8}\normalsize x+ \Large\frac{1}{2}= \Large\frac{1}{4}[/latex] [latex]\Large\frac{1}{8}\normalsize(\color{red}{-2})+ \Large\frac{1}{2}\normalsize\stackrel{\text{?}}{=} \Large\frac{1}{4}[/latex] [latex]\Large\frac{-2}{8}+ \Large\frac{1}{2}\normalsize\stackrel{\text{?}}{=} \Large\frac{1}{4}[/latex] [latex]\Large\frac{-2}{8}+ \Large\frac{4}{8}\normalsize\stackrel{\text{?}}{=} \Large\frac{1}{4}[/latex] [latex]\Large\frac{2}{8}\normalsize\stackrel{\text{?}}{=} \Large\frac{1}{4}[/latex] [latex]\Large\frac{1}{4}= \Large\frac{1}{4}\quad\checkmark[/latex]

In the example above, the least common denominator was [latex]8[/latex]. Now it’s your turn to find an LCD, and clear the fractions before you solve these linear equations.

Try it

Notice that once we cleared the equation of fractions, the equation was like those we learned how to solve earlier. We changed the problem to one we already knew how to solve!

Solve equations by clearing the Denominators

Find the least common denominator of all the fractions in the equation.
Multiply both sides of the equation by that LCD. This clears the fractions.
Isolate the variable terms on one side, and the constant terms on the other side.
Simplify both sides.
Use the multiplication or division property to make the coefficient on the variable equal to [latex]1[/latex].

Here’s an example where you have three variable terms. After you clear fractions with the LCD, you will simplify the three variable terms, then isolate the variable.

Example

Solve: [latex]7=\Large\frac{1}{2}\normalsize x+\Large\frac{3}{4}\normalsize x-\Large\frac{2}{3}\normalsize x[/latex]

Show Solution

Now here’s a similar problem for you to try. Clear the fractions, simplify, then solve.

Try it

Caution!

One of the most common mistakes when you clear fractions is forgetting to multiply BOTH sides of the equation by the LCD. If your answer doesn’t check, make sure you have multiplied both sides of the equation by the LCD.

In the next example, we’ll have variables and fractions on both sides of the equation. After you clear the fractions using the LCD, you will see that this equation is similar to ones with variables on both sides that we solved previously. Remember to choose a variable side and a constant side to help you organize your work.

Example

Solve: [latex]x+\Large\frac{1}{3}=\Large\frac{1}{6}\normalsize x-\Large\frac{1}{2}[/latex]

Show Solution

Now you can try solving an equation with fractions that has variables on both sides of the equal sign. The answer may be a fraction.

Try it

In the following video, we show another example of how to solve an equation that contains fractions and variables on both sides of the equal sign.

In the next example, we start with an equation where the variable term is locked up in parentheses and multiplied by a fraction. You can clear the fraction, or if you use the distributive property, it will eliminate the fraction. Can you see why?

EXAMPLE

Solve: [latex]1=\Large\frac{1}{2}\normalsize\left(4x+2\right)[/latex]

Show Solution

Now you can try solving an equation that has the variable term in parentheses that are multiplied by a fraction.

Try it

In the following video, we show how to solve a multi-step equation with fractions.

If you like to work with fractions, you can just apply your knowledge of operations with fractions and solve. Regardless of which method you use to solve equations containing variables, you will get the same answer. You can choose the method you find the easiest! Remember to check your answer by substituting your solution into the original equation.

Solving Equations By Clearing Decimals

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

Example

Solve [latex]3y+10.5=6.5+2.5y[/latex] by clearing the decimals in the equation first.

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. Note 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]\Large\frac{2}{3}\normalsize x=18[/latex]
Multiply by the reciprocal of [latex]\Large\frac{2}{3}[/latex] .	[latex]\Large\frac{\color{red}{3}}{\color{red}{2}}\normalsize\cdot\Large\frac{2}{3}\normalsize x[/latex]
Reciprocals multiply to one.	[latex]1x=\Large\frac{3}{2}\normalsize\cdot\Large\frac{18}{1}[/latex]
Multiply.	[latex]x=27[/latex]
Check your answer.	[latex]\Large\frac{2}{3}\normalsize x=18[/latex]
Let [latex]x=27[/latex].	[latex]\Large\frac{2}{3}\normalsize\cdot\color{red}{27}\stackrel{\text{?}}{=}18[/latex]
	[latex]18=18\quad\checkmark[/latex]

Find the least common denominator of all the fractions in the equation.	[latex]7=\Large\frac{1}{2}\normalsize x+\Large\frac{3}{4}\normalsize x-\Large\frac{2}{3}\normalsize x\quad{LCD=12}[/latex]
Multiply both sides of the equation by [latex]12[/latex].	[latex]\color{red}{12}(7)=\color{red}{12}\cdot\Large(\frac{1}{2}\normalsize x+\Large\frac{3}{4}\normalsize x-\Large\frac{2}{3}\normalsize x\Large)[/latex]
Distribute.	[latex]12(7)=12\cdot\Large\frac{1}{2}\normalsize x+12\cdot\Large\frac{3}{4}\normalsize x-12\cdot\Large\frac{2}{3}\normalsize x[/latex]
Simplify — and notice, no more fractions!	[latex]84=6x+9x-8x[/latex]
Combine like terms.	[latex]84=7x[/latex]
Divide by [latex]7[/latex].	[latex]\Large\frac{84}{\color{red}{7}}=\Large\frac{7x}{\color{red}{7}}[/latex]
Simplify.	[latex]12=x[/latex]
Check: Let [latex]x=12[/latex].
[latex]7=\Large\frac{1}{2}\normalsize x+ \Large\frac{3}{4}\normalsize x- \Large\frac{2}{3}\normalsize x[/latex] [latex]7\stackrel{\text{?}}{=} \Large\frac{1}{2}\normalsize(\color{red}{12})+ \Large\frac{3}{4}\normalsize(\color{red}{12})- \Large\frac{2}{3}\normalsize(\color{red}{12})[/latex] [latex]7\stackrel{\text{?}}{=}6+9-8[/latex] [latex]7=7\quad\checkmark[/latex]

Find the LCD of all the fractions in the equation.	[latex]x+\Large\frac{1}{3}=\Large\frac{1}{6}\normalsize x-\Large\frac{1}{2}\normalsize,\quad{LCD=6}[/latex]
Multiply both sides by the LCD.	[latex]\color{red}{6}(x+\Large\frac{1}{3}\normalsize)=\color{red}{6}(\Large\frac{1}{6}\normalsize x-\Large\frac{1}{2})[/latex]
Distribute.	[latex]6\cdot{x}+6\cdot\Large\frac{1}{3}\normalsize=6\cdot\Large\frac{1}{6}\normalsize x-6\cdot\Large\frac{1}{2}[/latex]
Simplify — no more fractions!	[latex]6x+2=x-3[/latex]
Subtract [latex]x[/latex] from both sides.	[latex]6x-\color{red}{x}+2=x-\color{red}{x}-3[/latex]
Simplify.	[latex]5x+2=-3[/latex]
Subtract 2 from both sides.	[latex]5x+2\color{red}{-2}=-3\color{red}{-2}[/latex]
Simplify.	[latex]5x=-5[/latex]
Divide by [latex]5[/latex].	[latex]\Large\frac{5x}{\color{red}{5}}=\Large\frac{-5}{\color{red}{5}}[/latex]
Simplify.	[latex]x=-1[/latex]
Check: Substitute [latex]x=-1[/latex].
[latex]x+\Large\frac{1}{3}= \Large\frac{1}{6}\normalsize x- \Large\frac{1}{2}[/latex] [latex](\color{red}{-1})+ \Large\frac{1}{3}\normalsize\stackrel{\text{?}}{=} \Large\frac{1}{6}\normalsize(\color{red}{-1})- \Large\frac{1}{2}[/latex] [latex](-1)+ \Large\frac{1}{3}\normalsize\stackrel{\text{?}}{=}- \Large\frac{1}{6}- \Large\frac{1}{2}[/latex] [latex]- \Large\frac{3}{3}+ \Large\frac{1}{3}\normalsize\stackrel{\text{?}}{=}- \Large\frac{1}{6}- \Large\frac{3}{6}[/latex] [latex]- \Large\frac{2}{3}\normalsize\stackrel{\text{?}}{=}- \Large\frac{4}{6}[/latex] [latex]- \Large\frac{2}{3}=- \Large\frac{2}{3}\quad\checkmark[/latex]

	[latex]1=\Large\frac{1}{2}\normalsize(4x+2)[/latex]
Distribute.	[latex]1=\Large\frac{1}{2}\normalsize\cdot4x+\Large\frac{1}{2}\normalsize\cdot2[/latex]
Simplify. Now there are no fractions to clear!	[latex]1=2x+1[/latex]
Subtract 1 from both sides.	[latex]1\color{red}{-1}=2x+1\color{red}{-1}[/latex]
Simplify.	[latex]0=2x[/latex]
Divide by [latex]2[/latex].	[latex]\Large\frac{0}{\color{red}{2}}=\Large\frac{2x}{\color{red}{2}}[/latex]
Simplify.	[latex]0=x[/latex]
Check: Let [latex]x=0[/latex].
[latex]1=\Large\frac{1}{2}\normalsize(4x+2)[/latex] [latex]1\stackrel{\text{?}}{=} \Large\frac{1}{2}\normalsize(4(\color{red}{0})+2)[/latex] [latex]1\stackrel{\text{?}}{=} \Large\frac{1}{2}\normalsize(2)[/latex] [latex]1\stackrel{\text{?}}{=} \Large\frac{2}{2}[/latex] [latex]1=1\quad\checkmark[/latex]

	[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]

Module 10: Linear Equations