Solving Equations Using the Least Common Multiple

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

There are several methods that can be used to solve equations. In this lesson, you'll learn about using the least common multiple method to help you combine and solve for your variable.

Least Common Multiple

You might remember from your fraction days that the least common multiple (LCM), the lowest number that is a multiple of all your given numbers, is a number that can help you combine fractions. For example, to add 1/2 and 3/4, you have to find the least common multiple for the denominator before you could add these two fractions together like this.

  • 1/2 + 3/4 = 2/4 + 3/4 = 5/4

The least common multiple for the denominator here is 4.

Using LCM to Solve Equations

Well, you can use the least common multiple to help you solve your algebraic equations when you need to combine fractions to help you simplify your equations. To do this, follow these steps.

  1. Move all your variable terms to one side of the equation and all your constant terms to the other.
  2. Use the LCM to combine any fraction terms that are like terms.
  3. Continue solving for your variable by using your algebra skills.

Practice Problem 1

Let's take a look at these steps in action now.

Let's solve this equation for the x variable.


solve with lcm


Following the steps, you first move all your variable terms to one side. So pick one side, say the left side. Now move all the constant terms to the other side, the right side. Here is what it looks like.


solve with lcm


Now, in the second step, combine your fractions with the help of the least common multiple. You are combining fractions, so your least common multiple will help you figure out what your denominators need to be before you can simplify this part of your equation.

You have 7 and 2 as denominators, so your least common multiple for these two numbers is 14. So you need to change both fractions so they both have a denominator of 14. Then you can combine them.


solve with lcm


After using the LCM to help you combine fraction terms, continue solving for your variable using your algebra skills.


solve with lcm


And you are done! Your answer is -28.

Practice Problem 2

Let's look at another example and see if you can solve it on your own before reading the answer that follows.

This time, solve this equation for the unknown variable.


solve with lcm


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