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Solving Rational Equations with Like Denominators - Problem 1

Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

When you come to solving equations that involve fractions or rational expressions, there’s lots of different ways to solve them. So I’m going to show you guys two different ways to solve this problem. I’m not sure which one you guys are going to use. If you watch a bright storm video where I showed you a trick about cancelling out the denominators, some of you guys might approach this problem like this.

You might say oh look those both have Xs in the denominator, so if I multiply this by x over x which is not changing the value I’m just multiplying by 1, then I could cancel out my denominators and just solve this problem. That’s one way to approach this problem which is excellent you’ll go through and you’ll get 3 equals -3x, or x equals 1. That’s totally fine. That’s one way to do this keeping in mind you can only cancel out the denominators if all of the denominators are the same. Don’t cancel them out and then just leave plain old 3 or -3 over here. I have to make sure that guy has x in the denominator as well.

Okay so that’s one way you guys might have done this problem. Let me show you another way which is more common. If I have 3x, 3 over x minus 6 over x equals -3, a lot of people will notice that the left hand side has a common denominator of x which is right. So I could combine fractions on the left here 3 take away 6 is -3 and work from here.

What I have here is problem I can do in my head. Some of you guys will look at this and know right away oh x must be 1 because of the -3s. If you didn’t recognize that right away, think of this as a proportion I have two equal fractions, if I cross multiply I’ll see -3x equals -3, x equals 1 that’s another way you can solve this problem. It’s totally up to you how you’re going to doing it, the important things are you can add or subtract fractions only if they have a common denominator, that’s the first important thing.

The other important thing is you can use the trick of eliminating the denominators only if all of your terms have that exact same denominator. In this problem I had to multiply by x over x in order to have the right hand side of my equation have that same denominator. One of the cool things about Math you guys is that you can do a problems in so many different ways. It’s totally up to you whatever works in your brain.

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