Lesson 1-5: Literal Equations
What type of equations?
So far we’ve talked about expressions and equations. Expressions are your basic math
sentence without an equal sign. An equation is a comparison of two expressions using
an equal size. So, what’s this literal equation stuff?
All of the expressions and equations we’ve played with so far have had just one
variable. Almost every time I ask you to give me a variable for an unknown quantity,
you’ve said x. It seems as if x is winning the popularity contest for variables.
It just so happens there is plenty of other variables out there; perfectly good variables
too! They all want in on the fun! What if one of them (let’s say y) butted in on an
equation of x’s. Yup, y just jumped right into the equation, right along with x. We now
have an equation with more than one variable! That is what we call a literal equation.
Where did that phrase come from? Well, the best I can tell is it comes from the Latin
word for “letter”…literal equations are “letter” equations…more than one letter/variable.
What can you do with a literal equation?
The main thing is you can see how all the variables relate to each other. You do that by
solving it for one of them. You can solve it for any of the variables in the equation. Just
pick one, do the algebra to get it all by itself and you have it! You’ll end up with the
variable you solved for all by its lonesome on one side, with all the other variables and
numbers on the other side.
Yeah, but all those letters confuse me!
No worries! Just try to think of the other variables as numbers…and do the algebra. I
know it can look confusing. Let’s try an example ok?
One way that we run into literal equations is when we are working with formulas.
Think about the formula for the area of a triangle: A 
1
bh . That looks familiar right?
2
Ok, how many letters (as opposed to numbers) does it have? That’s right, 3: A, b, and
h. What do those letters stand for? A is the area, b is the length of the base, and h is
the height. Cool, you got it! This is an example of a literal equation.
Now, can you solve that formula for b the base? Sure…just treat the other variables A
and h like numbers and work to get b all by itself. Let’s do it together:
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Lesson 1-5: Literal Equations
1
A  bh
2
2 A  bh
2 A bh

h
h
2A
b
h
1
first...multiply both sidesby 2
2
Now get rid of theh...divideboth sidesby h
Get bby itself ...get rid of the
...and cancel
There you have it! Let’s try another…
Example – the Fahrenheit/Celsius formula
The formula to convert Celsius temperature to Fahrenheit is F 
9
C  32 . If you know
5
the temperature in Celsius, you plug it into this formula to see what it is in Fahrenheit.
Let’s convert this formula so that if you know Fahrenheit you can see what it is in
Celsius. That means we need to solve the above formula for … drum roll please … C.
9
F  C  32
5
9
F  32  C  32  32
5
5
9
5
( F  32)  ( C )
9
5
9
C  5 ( F  32)
9
Get C by itself ...
...subtract 32 fromboth sides
9
5
Now get rid of the ...multiply both sides by reciprocal
Cancel and you haveit !
Example – the surface area of a cylinder
The formula to calculate the surface area of a cylinder is A  2 rh where r is the radius
and h is the height. If you knew the area and height, you could find out what the radius
is. Solve for r to get the general formula:
A  2 rh
A
2  rh

2 h 2  h
A
r
2 h
Sinceeverything withthe r is combined with multiplication...
...get rid of it inone step
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Lesson 1-5: Literal Equations
Example – a bit of a tricky one!
Solve the following for h:
b 3
 . What makes this tricky? The variable you want to
h 4
solve for is in the denominator. We need it to be all by itself, i.e. not in a fraction. How
can we move it out? Let me answer that with a question (you hate that don’t you):
x
 1 ? You multiply both sides by the
3
reciprocal which is 3. After you do so, you have x  3 … look at that. Basically what you
What do you do to get rid of the 3 in this:
did was move the 3 from the denominator of one side to the numerator of the other side!
So back to our original problem:
b 3

h 4
h b
3 h
( )( )
1 h
4 1
3h
b
4
4
3h 4
b  ( )
3
4 3
4b
h
3
Get h byout of thedenominator...
...multiply both sides byits reciprocal
Now get rid of
3
...multiply both sides byits reciprocal
4
Cancel...
...and we're done!
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