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.
P 4s
=
4
4
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.
P
=s
4
It is customary to write the resulting literal
equation with the desired variable placed
first or:
P
s=
4
Let’s look at another example:
c+d
Solve: q =
for “c” :
2
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!
c+d
q=
2
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:
(
c+d
c +d)
q=
⇒q=
2
2
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.
(
c +d)
q=
2
(
c +d)
2q =
2
2
2q = c + d
2q - d = c + d - d
2q − d = c
c = 2q − d