In this video, we are going to solve fractional equations, such as \dfrac{4}{x} + \dfrac{2}{6} = \dfrac{5}{x}, by using the least common denominator. After you finish this lesson, view all of our Algebra 1 lessons and practice problems.

For example: \dfrac{4}{x}+\dfrac{2}{6}=\dfrac{5}{x}, find the value of x

First, find the least common denominator, which is 6x. Multiple every numerator by 6x
\dfrac{(6x)4}{x}+\dfrac{(6x)2}{6}=\dfrac{(6x)5}{x}

The x and 6 cancel out with the denominator, leaving us with
(6)4+(x)2=(6)5
24+2x=30

Now, let’s solve it like any other equations. Subtract 24 from both sides.
2x=6

Isolate the x and we have
x=3

Fractional Equations

Examples of Fractional Equations

Example 1

\dfrac{2}{x}+\dfrac{1}{4}=\dfrac{5}{3x}

First, find the least common denominator, which is 12x. Multiple every numerator by 12x

\dfrac{12x(2)}{x}+\dfrac{12x(1)}{4}=\dfrac{12x(5)}{3x}

The x, 4, and 3x cancel out with the denominator, leaving us with

12(2)+3x(1)=4(5)

24+3x=20
Subtract 24 from both sides

24+3x-24=20-24

3x=-4
Divide 3 on both sides to isolate x
\dfrac{3x}{3}=\dfrac{-4}{3}
Now, we have
x=\dfrac{-4}{3}

Example 2

\dfrac{3}{4}-\dfrac{5m}{4}=\dfrac{108}{24}

First, find the least common denominator, which is 24. Multiple every numerator by 24

\dfrac{24(3)}{4}-\dfrac{24(5m)}{4}=\dfrac{24(108)}{24}

The 4 and 24, cancel out with the denominator, leaving us with

6(3)-6(5m)=108

18-30m=108
Subtract 18 from both sides

18-30m-18=108-18

-30m=90
Divide -30 on both sides to isolate x
\dfrac{-30m}{-30}=\dfrac{90}{-30}
Now, we have
m=-3

Video-Lesson Transcript

In this lesson, we’ll go over fractional equations.

This is just an equation involving fractions. But it’s algebra.

So, it’s going to be a little complicated.

For example:

\dfrac{4}{x} + \dfrac{2}{6} = \dfrac{5}{x}

Let’s go back to the regular fractions for a second. Just a little side note.

If we’re going to add

\dfrac{1}{2} + \dfrac{1}{3}

we need a common denominator.

We can’t just add it the way it is.

The common denominator for these is 6.

In order to change the denominator by 6, we have to multiply the numerator and the denominator by the same number.

In the first fraction, we have to multiply by 3 and the second fraction has to be multiplied by 2.

= \dfrac{3 \times 1}{3 \times 2} + \dfrac{1 \times 2}{3 \times 2}
= \dfrac{3}{6} + \dfrac{2}{6}

We’re going to use the same concept in solving fractional equations.

First, we have to find the least common multiple of the denominators.

So our least common denominator (LCD) is 6x.

But instead of getting the least common denominator and still having fractions, we’ll just multiply everything by the least common denominator.

If we multiply everything by 6x, things are going to get canceled out.

And it’s not going to be a fractional equation anymore. It’s going to be regular equations.

\dfrac{4}{x} + \dfrac{2}{6} = \dfrac{5}{x}
(6x) \dfrac{4}{x} + (6x) \dfrac{2}{6} = (6x) \dfrac{5}{x}

Now, we can cancel the denominators.

24 + 2x = 30

Now, we can solve this like a regular equation.

We want to isolate x using inverse operations.

24 - 24 + 2x = 30 - 24
2x = 6
\dfrac{2x}{2} = \dfrac{6}{2}

Our answer is

x = 3

So, for fractional equations, we have to find the least common denominator.

But we’re not going to manipulate them to have a common denominator.

What we’re going to do instead is to multiply each term by the least common denominator so that the denominator of each term cancels out.

Then we end up with a new equation where we can solve for x.