Solving Literal Equations

Sometimes you have a formula and you need to solve for some variable other than the "standard" one. Example: Perimeter of a square P = 4s It may be that you need to solve this equation for s, so you can plug in a perimeter and figure out the side length.

This process of solving a formula for a given variable is called "solving literal equations".

One of the dictionary definitions of "literal" is "related to or being comprised of letters“. Variables are sometimes referred to as literals.

So "solving literal equations" may just be another way of saying "taking an equation with lots of variables, and solving for one variable in particular.”

To solve literal equations, you do what you've done all along to solve equations, except that, due to all the variables, you won't necessarily be able to simplify your answers as much as you're used to doing.

Here's how "solving literal equations" works: Suppose you wanted to take the formula for the perimeter of a square and solve it for “s” (or the length of the side) instead of using it to solve for perimeter.

P=4s

Just as when we were solving linear equations, we want to 1.) identify the variable we want to solve for. In this case the length of the side, “s”. Next, we want to 2.) isolate the variable. What we mean, is get it on one side of the equation or the other, by itself

Now, that does not mean we move things in any old fashion, like simply moving the “4” to the other side to give 4P = s, THIS WOULD BE WRONG. Instead, we need to ask ourselves what operation is being done on the variable we want to isolate. In this case the variable “s” is being multiplied by “4”

Once we determine the operation being done on the variable we want to isolate, we apply the reverse operation to undo it. For example, since the variable we want, “s” is being multiplied by “4”, we must divide by “4” because division is the reverse of multiplication. We must apply this to the other side of the equation to maintain

equality.

44

4sP

=

Notice that on the right, the “4” in the numerator is cancelled (undone) by the “4” in the denominator, resulting in the variable “s” having been isolated.

sP=

4It is customary to write the resulting literal equation with the desired variable placed first or:

4Ps =

Let’s look at another example:

Solve: 2

dcq += for “c” :

Step 1: Identify the variable to solve for. Here that variable is c Step 2: Isolate this variable. Determine what operation(s) is (are)

being performed on the variable. In this case, “d” is being added to “c” and “c” is being divided by “2”

Now, which one of the two reverse operations do we do first? Do we multiply by “2” or subtract “d”? The rule is easy, just go in reverse order of operations. Order of Operations (OoO) has us do items in parenthesis first, followed by powers and roots, followed by multiplication and division, followed finally by addition and subtraction.

So to solve, we apply a reverse order of operations (ROoO), removing variables that are added or subtracted first, then those being multiplied or divided and so on. Let’s look again at the original equation!

2dcq +

=

An initial glance appears as though we should start by subtracting d from both sides since in ROoO, subtracting would precede multiplying.

This would unfortunately be wrong. The reason is that the original equation can be rewritten as: ( )

22dcqdcq +

=⇒+

=

This grouping is implied by the line that is under both “c” and “d”. As a result, when using ROoO, we undo the division by “2” first by multiplying by “2” and then undo the grouped “d” by subtracting it.

( )2

dcq +=

dcq +=2

( )2

dcq +=2 2

dcq += 2 - d - d

cdq =−2

dqc −= 2