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Solving Formulas - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Solving formulas for a variable is a critical skill in the Geometry area unit because many problems will give the area of a polygon and ask for a side, height, or some other dimension. In these cases, simply substituting and typing into a calculator will not yield the correct answer. The successful Geometry student must be capable of substituting into a formula and then solving formulas for the one remaining variable.

When you are in teh area unit, you have to be adept at solving formulas. So what is a formula? Well, in a formula is an equation where you have multiple variables.
So this equation says the area of a rectangle is equal to its base times its height. So you have to be able to look at this equation and solve for different variables. So I'm going to solve this once for b and once for h just to show you different ways of manipulating a formula. Right now it's isolated for a.
So if I'm trying to solve for b, first thing I ask myself is what is happening to that variable that I'm solving for. Am I adding something to it? Am I dividing something? Well, it looks like I'm multiplying it by h. Remember when you have two variables next to each other, the implied operation is multiplication. So how do you undo multiplication?
Well, the opposite of multiplying is dividing. So if I divide by h. Again since I'm trying to isolate b and if I divide by h on the other side, because we have an equal sign, you have to do the same operation to both sides. Each divided by h is 1, so I'm going to write a big 1 there. So we just have b on one side and on the other side a and h can't divide those 2. So it's just going to be a divided by h. So notice that we have isolated b in one step.
Same equation. Let's say we're solving for h. So, if I look at h, it's being multiplied by b. So I'm going to undo multiplying by dividing. So b divided by b is 1, so we find that h is a divided by b.
If we move on to something that has division and multiplication, the triangle formula. Let's say we want to solve this for b. So if I'm looking at b, I'm being multiplied by h and I'm being divided by 2.
So you could do this in whatever order that you want. I think it's easiest to start off by eliminating your fraction. So I'm going to multiply both sides by 2. Since we're dividing by 2, I said to myself that the opposite dividing is multiplying.
So we have 2 times a is equal to b times h. I have not finished solving for b because I have to divide by h. So base is equal to 2 times the area divided by the height. Okay?
Same equation, let's say I want to solve it for h. Well, I'm going to do the same first step which is multiplying by 2 thereby eliminating our fraction. So we have 2a equals b times h, and since I'm solving now for h, I'm going to get rid of the b. So I'm going to divide both sides by b and h is equal to 2 times a divided by b.
Key to solving formulas is always identifying your variable and then asking yourself what are you doing to that variable so you can undo it.

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