# Solve linear formulas for a given variable

Solving formulas is much like solving general linear equations. The only difference is we will have several variables in the problem and we will be attempting to solve for one specific variable. For example, we may have a formula such as $A=\phi r^2+\phi rs$ (formula for surface area of a right circular cone) and we may be interested in solving for the variable s. This means we want to isolate the s so the equation has s on one side, and everything else on the other. So a solution might look like $s=\frac{A-\phi r^2}{\phi s}$. This second equation gives the same information as the first, they are algebraically equivalent, however, one is solved for the area, while the other is solved for s (slant height of the cone). In this section we will discuss how we can move from the first equation to the second.

When solving formulas for a variable we need to focus on the one variable we are trying to solve for, all the others are treated just like numbers. This is shown in the following example. Two parallel problems are shown, the first is a normal one step equation, the second is a formula that we are solving for x.

Example 1:

Solving for the x solution

3x = 12 and wx = z

In both problems, x is multiplied by something, To isolate the x we divide by 3 or w.

Our solution x = 4 and $x=\frac{z}{w}$.

We use the same process to solve 3x = 12 for x as we use to solve wx = z for x. Because we are solving for x we treat all the other variables the same way we would treat numbers. Thus, to get rid of the multiplication we divided by w. This same idea is seen in the following example.

Example 2:

Solve the solution m + n = p for n

Solving for n, treat all other variables like numbers

Subtract m from both sides

So the solution is n = p – m

As p and m are not like terms, they cannot be combined. For this reason we leave the expression as p – m. This same one-step process can be used with grouping symbols.

Example 3:

Find the solution a(x – y) = b for a

for a Solving for a, treat (x – y) like a number

Divide both sides by (x – y)

$\frac{a(x-y)}{x-y}=\frac{b}{x-y}$

So the solution is $a=\frac{b}{x-y}$

Because (x – y) is in parenthesis, if we are not searching for what is inside the parenthesis, we can keep them together as a group and divide by that group. However, if we are searching for what is inside the parenthesis, we will have to break up the parenthesis by distributing. The following example is the same formula, but this time we will solve for x.

Example 4:

Find the solution a(x – y) = b for x

Solving for x, we need to distribute to clear parenthesis

ax – ay = b

This is a two -step equation, ay is subtracted from our x term, Add ay to both sides

ax = b + ay

The x is multiplied by a, Divide both sides by a

So the solution is $x=\frac{b+ay}{a}$.

Be very careful as we isolate x that we do not try and cancel the a on top and bottom of the fraction. This is not allowed if there is any adding or subtracting in the fraction. There is no reducing possible in this problem, so our final reduced answer remains $x=\frac{b+ay}{a}$. The next example is another two-step problem

Example 5:

Solve y = mx + b for m

$\frac{y-b}{x}=m$