Solving Literal Equations

This is the formula for the area A of a rectangle with base b and height h. They're asking me to solve this formula for the base b.

If they'd asked me to solve 3 = 2b for b, I'd have divided both sides by 2 in order to isolate (that is, in order to get by itself, or solve for) the variable b. I'd end up with the variable b being equal to a fractional number.

In this case, I won't be able to get a simple numerical value for my answer, but I can proceed in the same way, using the same step for the same reason (namely, that it gets b by itself). So, following the same reasoning for solving this literal equation as I would have for the similar one-variable linear equation, I divide through by the "h":

This equation is the "uniform rate" equation, "(distance) equals (rate) times (time)", that is used in "distance" word problems, and solving this for the specified variable works just like solving the previous equation.

The variable they want has a letter multiplied on it; to isolate the variable, I have to divide off that letter. So I'll solve for the specified variable r by dividing through by the t:

This is the formula for the perimeter P of a rectangle with length L and width w.

If they'd asked me to solve 3 = 2 + 2w for w, I'd have subtracted the "free" 2 over to the left-hand side, and then divided through by the 2 that's multiplied on the variable. I can follow the exact same steps for this equation:

The variable I need to isolate is currently inside a fraction. I need to get rid of the denominator. To do this, I'll multiply through by the denominator's value of 2. On the left-hand side, I'll just do the simple multiplication. On the right-hand side, to help me keep things straight, I'll convert the 2 into its fractional form of 2/1.

If they'd asked me to solve for t, I'd have multiplied through by t, and then divided both sides by 5. Following the same reasoning and doing the same steps, I get:

I need to get the variable a by itself. Currently, it's multiplied onto other stuff in two different terms. I can't combine those terms, because they have different variable parts. (The first term has no other variable, but the second term also has the variable c.)

I want to divide off the stuff that's multiplied on the specified variable a, but I can't yet, because there's different stuff multiplied on it in the two different places. But what if I factor the a out front?

Since each of the two fractions on the right-hand side has the same denominator of 2, I'll start by multiplying through by 2 to clear the fractions. Then I'll work toward isolating the variable h.

This is a big, lumpy equation, but the solution method is the same as always. The variable I want has some other stuff multiplied onto it and divided into it; I'll divide and multiply through, respectively, to isolate what I need.