Look at you go! You’ve now learned how to solve one-step equations, two-step equations, and multi-step equations.
Now it’s time to discuss solving literal equations.
In this lesson, we will define literal equations, examine examples of literal equations, as well as learn how to solve literal equations. Let’s literally get excited to begin!
What We Review
What is a literal equation?
Recall from our earlier study, that an equation is a mathematical sentence that uses an equal sign, = , to show that two expressions are equal. Unlike other equations you have already worked with, literal equations are equations primarily made up of letters and variables.
Many of the literal equations you have worked with in your life have been formulas. Although these equations will look different than our normal equations, they still follow the same rules of solving.
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Examples of literal equations
Although the idea of equations including mostly letters may seem like a foreign topic, you have used literal equations many times in your life. Here are some of the examples of literal equations you have already worked with in your life:
Area of a Rectangle
A = b \cdot h
Circumference of a Circle
C = \pi \cdot D
Simple Interest Formula
I = p \cdot r \cdot t
Each letter (or variable) in the literal equation has a specific meaning and changes from problem to problem.
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How to solve literal equations
Solving literal equations follows the same rules as solving a one-step or two-step equation. The idea of “solving” a literal equation essentially means we are rearranging the letters (or variables) to isolate a new variable. A literal equation is “solved” when the variable of interest is alone on one side of the equation.
Similar to solving equations, we will use inverse operations to isolate the variable by itself. Here are examples of inverse operations:
\text{Addition} \leftrightarrow \text{Subtraction}
\text{Multiplication} \leftrightarrow \text{Division}
Here are a few examples of solving literal equations:
Example 1
Solve for h in the following literal equation:
A = b \cdot h
Remember this formula? As stated above, this is the Area of a Rectangle. As noted previously, there are mainly letters and variables in Literal Equations. If this was a simple equation such as 10 = 2x , we would’ve simply divided both sides by 2 to achieve my final answer.
When “solving” Literal Equations, we follow the same rules as simple equations. Therefore, in order to solve for h in this equation, we need to isolate it by itself. Therefore, we will divide both sides by b .
\dfrac{A}{b} = \dfrac{b \cdot h}{b}
Doing so will isolate h , giving us the answer:
h = \dfrac{A}{b}
Example 2:
Although formulas are a common example of Literal Equations, not all Literal Equations are formulas. We can also rearrange and “solve” a Literal Equation for any variable. For Example:
Solve for m in the following equation:
| x = m + n | Original equation |
| x \textcolor{red}{- n} = m + n \textcolor{red}{- n} | Subtract n from both sides |
| x - n = m | m is now isolated |
| m = x - n |
Even though there were no numbers in the equation, we have “solved” the Literal Equation for m .
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Multi-step literal equations
Example 1
Not all Literal Equations only take one step to solve. Here is an example of using multiple steps to solve Literal Equations.
Solve for r in the following equation:
V = \pi r^2 h
Given is the formula for the Volume of a Cylinder. In order to solve for r , we must first get r^2 by itself:
\dfrac{V}{\pi h} = \dfrac{\pi r^2 h}{\pi h}
Then we are left with:
\dfrac{V}{\pi h} = r^2
Then to solve for r , we must take the square root of both sides:
\sqrt{\dfrac{V}{\pi h}} = \sqrt{r^2}
Now we have r isolated by itself, giving us the new Literal Equation:
r = \sqrt{\dfrac{V}{\pi h}}
Example 2
Here is an example of a Literal Equation that is not a formula, but which we can still solve for a variable.
Solve for x in the following equation:
4(x + y) = P
There are two ways to approach this problem. The first method is to treat it as an equation and distribute the 4 , then solve:
4x + 4y = P
We can then subtract 4y from each side:
4x + 4y \textcolor{red}{- 4y} = P \textcolor{red}{ - 4y}
Then we will need to divide each side by 4 :
\dfrac{4x}{4} = \dfrac{P - 4y}{4}
Finally, we need to simplify our equation:
x = \dfrac{P}{4} - \dfrac{4y}{4}
x = \dfrac{P}{4} - y
Now, we have finally solved for x in the Literal Equation. Let’s see how you can solve the equation, without having to simplify at the end.
In the other method, we can simply divide by 4 at the beginning to avoid using the Distributive Property. For example:
\dfrac{4(x + y)}{4} = \dfrac{P}{4}
Then, we would simply have to subtract y from each side:
x + y \textcolor{red}{- y} = \dfrac{P}{4} \textcolor{red}{ - y}
Therefore, we end up with:
x = \dfrac{P}{4}- y
Notice how the equation is already simplified, and no other steps are needed.
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Here’s a short video demonstrating how to solve literal equations:
Literal equations with fractions
Let’s work on some examples of literal equations involving fractions!
Example 1
Many literal equations, and formulas, involve fractions in some sort. For example, here’s the formula for the Volume of a Sphere:
V = \dfrac{4}{3} \pi r^3
Let’s say we are asked to solve for the radius, r . First, we must eliminate the fraction. Let’s do this by multiplying by the reciprocal.
| V = \dfrac{4}{3} \pi r^3 | Original equation |
| \dfrac{3}{4} \cdot V = \dfrac{3}{4} \cdot \dfrac{4}{3} \pi r^3 | |
| \dfrac{3}{4} \cdot V = \pi r^3 | |
| \dfrac{3V}{4 \textcolor{red}{\pi}} = \dfrac{\pi r^3}{\textcolor{red}{\pi}} | Divide each side by \pi |
| \dfrac{3V}{4\pi} = r^3 | Simplify |
| \sqrt[3]{\dfrac{3V}{4\pi}} = \sqrt[3]{r^3} | Cube Root both Sides |
| r = \sqrt[3]{\dfrac{3V}{4\pi}} |
Now that r is isolated, we have successfully solved for r .
Example 2
What happens if we simply want to rearrange an equation for the other variable?
For instance, solve the following equation for x :
y = \dfrac{x}{4} - \dfrac{1}{8}
Notice how there are two variables in the equation and our ultimate goal is still to isolate x . Remember, we can eliminate all fractions in one move by multiplying all terms by the Least Common Denominator.
In this Literal Equation, the least common denominator is 8 . Therefore, we will multiply each term by 8 .
8 \cdot y = 8 \cdot \dfrac{x}{4} - 8 \cdot \dfrac{1}{8}
This will give us an equation that no longer has fractions:
8y = 2x - 1
Then continue solving just as you would a normal equation:
| 8y = 2x - 1 | Original equation |
| 8y \textcolor{red}{+ 1} = 2x - 1 \textcolor{red}{+ 1} | Add 1 to each side |
| 8y + 1 = 2x | Simplify |
| \dfrac{8y + 1}{\textcolor{red}{2}} = \dfrac{2x}{\textcolor{red}{2}} | Divide each side by 2 |
| \dfrac{8y}{2} + \dfrac{1}{2} = x | Simplify |
| 4y + \dfrac{1}{2} = x | Simplify |
Now that we have isolated x by itself, we have correctly “solved” the Literal Equation.
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Keys to Remember: Solving Literal Equations
- A Literal Equation is an equation that contains all letters (or variables) or an Equation that has multiple variables
- Formulas, such as P = 2L \cdot 2W , are common examples of Literal Equations
- We solve Literal Equations by isolating a determined variable on one side of the equation
- Solving Literal Equations follow the same rules as normal equations, so we must do Inverse Operations in order to solve
- Remember, whatever you do to one side, you must do to the other!
- To solve Literal Equations with fractions, you can multiply each term by the Least Common Denominator to eliminate the fractions
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