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 $x=3$ ## 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. $\dfrac{1}{2} + \dfrac{1}{3}$

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. $24 - 24 + 2x = 30 - 24$ $2x = 6$ $\dfrac{2x}{2} = \dfrac{6}{2}$ $x = 3$
Then we end up with a new equation where we can solve for $x$.