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Solving Literal Equations

Literal equations, simply put, are equations containing two or more variables. Your goal is to solve for just one variable with respect to others. If you know how to solve regular equations, then I guarantee you that solving literal equations will be a breeze.

The "heart" of literal equation is to isolate or keep by itself a certain variable on one side of the equation (either left or right) and the rest on the opposite side. So, the key idea looks like this...

diagram literal equation

 

Let's go over some examples!

 

Example 1: Solve for s in the literal equation P = 4s.

Remember this formula? This is the perimeter of a square where P stands for perimeter and s stands for side. To solve for s, we need to get rid of the coefficient 4 which is multiplying s.  The inverse of multiplication is division, and that's why we should divide both sides by 4!

 

P=4s   We can isolate variable s on the right side.
P/4=4s/4   Divide both sides by 4
P/4=s   Simplify, now variable s stands by itself - done!

 

 

Example 2: Solve for L in the literal equation P = 2L + 2W.

 

Well, this is the formula to get the perimeter of a rectangle where: P = perimeter, L = length, and W = width. It is possible to isolate the variable L on the right side, however, why not flip the equation around so we can keep the variable L alone on the left?  Sounds like a plan!

Don't be intimidated by how it looks. Just focus on the things you want to do, that is to solve for L and the rest of the steps will follow.

 

P=2L+2W  

We want L solved, right?

2L+2W=P  

Flip around the equation to isolate variable on the left side.

2L+2W-2W=P-2W  

Subtract both sides by 2W

2L=P-2W  

Simplify

2L/2=(P-2W)/2  

Divide both sides by 2

L=(P-2W)/2  

Simplify, now L stands alone - solved!

 

 

Example 3: Solve for x in the literal equation m=(x-y)/3

What makes this literal equation interesting is that we are going to isolate a variable that is part of the numerator of a fraction. I'm not sure if you remember that whenever you see something like this, try to get rid of the denominator first. This makes the entire solving process a lot simpler.

Since that denominator 3 is dividing the expression "x − y", the opposite operation than can undo it is multiplication. It makes sense to multiply both sides by 3 first, then add by "y" to keep the x by itself. Not too bad, right?

 

m=(x-y)/3  

Okay, we want to solve for x. Let's isolate it on the right side.

3*m=[(x-y)/3]*3  

Start by multiplying both sides by 3 which is the denominator of the fraction.

3m=x-y  

Simplify. That's good, the denominator is now gone.

3m+y=x-y+y  

Add both sides by y. That's the only way to eliminate that − y on the right side.

3m+y=x  

Simplify, x is now happily by itself. Done!

 

 

Example 4: Solve for C in the literal equation F=(9C/5)+32

thermometer  

This is the formula used to convert the measure of temperature in Celsius unit to Fahrenheit scale. Notice that to find the value of F (Fahrenheit), we need to plug in some value of C (Celsius).

However, can we also use the given formula to find Celsius whenever the value for Fahrenheit is given?

Absolutely yes!

This is a literal equation after all so it is possible to express C in terms of F. That's what we're going to do now...

 

F=(9C/5)+32   All eyes on C . The goal is to isolate it.
F-32=(9C/5)+32-32   Well, get rid of that 32 on the right, by subtracting both sides by 32
F-32=(9C/5)   This looks cleaner after simplification.
5*(F-32)=(9C/5)*5   Next, multiply both sides by 5 to cancel out that denominator 5 under 9C.
5(F-32)=9C   We're getting there! I suggest not to distribute that 5 into (F − 32).
[5(F-32)]/9=9C/9   One more step, divide both sides by 9 to finally isolate the variable on the right.
[5(F-32)]/9=C   That's it! We have solved for C.

 

Example 5: Solve for h in the literal equation 3h + g = 5h − hg

 

This is really interesting! Some of you might think that it is impossible to isolate the variable h since it is found pretty much in three places - one h on the left and two on the right.  Well, don't give up yet! Let me show you a little "secret".

Use factoring method to pick that variable out of the group. But before you could factor h out, make sure that you move all the h's on one side of the equation.

Since we have two terms of h's on the right side, we might as well move the term 3h on the left to the other side.

 

3h+g=5h-hg   We want h isolated, right? 
3h-3h+g=5h-3h-hg   Keep all our h terms on the right side. We can do that by subtracting both sides by 3h.
g=2h-hg  

After simplification, it's wonderful to see all our h terms just on the right side.

It's obvious that the step should involve factoring h out.

g=h(2-g)  

Wow, this is great! Just a single h on the right side.

To isolate h by itself, that implies that we have to get rid of the expression (2 − g).

g/(2-g)=[h(2-g)]/(2-g)   Divide both sides by (2 − g)
cancel (2-g) on the right side   Doing some cancellation on the right side.
g/(2-g)=h   That is it! We have solved for h.

 

Example 6: Solve for x in the literal equation (x-2)/(3y-5)=(x/3)

 

The most straightforward way of solving this literal equation is to perform cross multiplication. In doing so, the denominators in both sides of the equation should disappear.

From that point, we can apply the same strategy from Example 5 to solve for x which involves gathering all x terms on one side of the equation and then hopefully factor the x out.

(x-2)/(3y-5)=(x/3)   In this equation we have two x's in both sides of the equation. More importantly, they are located in the numerator position.
3(x-2)=x(3y-5)   We want the denominators gone so without any hesitation we should apply the cross multiplication technique.
3x-6=3xy-5x  

Then simply apply distributive property in both sides of the equation. If you forgot how to do it, check it out in this lesson.

3x+5x-6=3xy-5x+5x  

At this point, we decide where to keep or gather all our x's. For this example, let's keep them on the left side.

Start by getting rid of that -5x on the right by adding both sides by 5x.

8x-6=3xy   This is how it looks after simplification. Next is to deal with that 3xy on the right side. Move that to the left as well.
8x-3xy-6=3xy-3xy   Subtract both sides by 3xy. That should keep all our x's on the left.
8x-3xy-6=0   Don't forget writing that 0 on the right side!
8x-3xy-6+6=0+6   Now that −6 on the left must be moved to the right side by adding both sides by 6.
8x-3xy=6   This is getting nicer! We have all our x terms on the left. It appears that we can factor that x out.
x(8-3y)=6   Yep! We just did!
[x(8-3y)]/(8-3y)=6/(8-3y)   Finally to solve x, we should divide both sides by the expression (8 − 3y). Perform some cancellation.
x=6/(8-3y)   And we're done!

 

 


 

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