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Solving Single-step Equations - Problem 4
Solving an equation and finding the value of the variable requires "undoing" what has been done to the variable. We do this by using inverse operations to isolate the variable. Remember that an equation is two expressions that are equal to each other. This means that when using inverse operations to isolate the variable, what is done to one side of the equation has to be done to the other side as well so that the equation stays balanced. In an equation, if "x" is in the denominator, start by multiplying both sides by x. This way, the "x" gets cancelled out from the denominator. Now you can solve the rest of the equation by continuing to use inverse operations. Make sure to keep in mind that any operation you do to one side of the equation, you must do to the other side as well. After solving for the variable, check your answer by plugging the value into the original equation. If the equation is true -- meaning the left side of the equation equals the right side of the equation -- then the solution is correct.
One of the more difficult solving equations problems that you'll see in single step equations is when x is in the denominator of the fraction and it's weird because not only do you have a fraction but x is in the bottom. But don't freak out, what this means is that 90 divided by some number gives you the answer 12. And the way you can get that x by itself is by undoing what's happening to it now.
So right now 90 is dividing x, I'm going to do the opposite which is multiplying both sides by x/1. So now those Xs cancel out and I'm left with 12x equals 90. That's an equation that I'm a lot more comfortable with because now there's no fraction going on, there's no division. To get x all by itself I need to undo that 12 times x business so now I have x equals 90/12 and depending on your textbook and your teacher that might be the appropriate final answer. A lot of textbooks say use your calculator to simplify or reduce all fractions. Since I'm a Math teacher I would make my students reduce that fraction. I want to show you how to do that.
First thing you do is in your brain you think about what number multiplies into 12 and into 90 and there's a couple of right answers. The first one that came to my brain was 3, so I'm going to divide both sides by 3 and I'll have, excuse me, divide top and bottom by three. That will give me 30/4 which again is not the most simple answer because there's still another number that goes into 30 and 4. That number is 2 so I need to divide by 2/2 and I'll get 15/2. That's my fractional answer that's most simplified. You might grab a calculator and turn that into 7.5. It's totally up to you and your teacher what form you leave it in.
Before we move on please make sure you check your solution. So go back to the original problem where we had, let's see 12 equals 90 over 15/2. That's ugly but don't freak out when you're dividing fractions that's the same thing as multiplying by the reciprocal right? So on this side I could simplify by doing 90 times the reciprocal. 12 is going to be equal to 90 times the reciprocal, so that's 2 on top of 15. I'm hoping 12 is equal to 180 divided by 15 and then if I'm right I'll be a happy camper. Let's just double check 180 divided by 15 you do indeed get 12. So that's how I know that I got 15/2 as the correct answer.
So the key to this problem, this type of problem where you have x in the denominator, is to do the opposite which is multiplying both sides by x.