Literal Equations
Learn how to solve equations for a specific variable when multiple variables are involved.
Definition
- Area of rectangle:
- Distance formula:
- Perimeter:
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Worked Examples
The area formula for a rectangle is . Solve for .
Identify what we want to isolate
We need to get by itself → Goal:
Identify what operation connects to other terms
and are multiplied together →
Apply the inverse operation to both sides
Divide both sides by →
Simplify
The 's cancel on the right side →
Answer:
Common Mistakes
Only applying an operation to one side of the equation
Why it's wrong: Whatever you do to one side MUST be done to the other side to maintain equality.
Correct: If you divide the left side by , you must also divide the right side by .
Trying to cancel incorrectly: thinking simplifies to
Why it's wrong: When dividing a sum, you must divide EVERY term, not just the first one.
Correct:
Forgetting to apply the reciprocal when clearing fractions
Why it's wrong: To undo multiplication by , you multiply by , not divide by then multiply by .
Correct: Multiply by the reciprocal:
Not identifying the correct variable to isolate
Why it's wrong: Reading the problem carefully is essential - different scenarios require solving for different variables.
Correct: Always identify which variable you need BEFORE you start solving.
Interactive Visual
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Balance Scale
Solution: x = 4
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Practice Problems
17 problemsWhich of the following is a literal equation?
Why It Matters
- Physics: Rearranging to find mass:
- Finance: Converting interest formulas to find principal, rate, or time
- Chemistry: Solving gas laws for different variables
- Construction: Finding missing dimensions from area or perimeter formulas
Real World Applications
Physics: Force and Motion
Newton's second law $F = ma$ relates force, mass, and acceleration. Scientists rearrange this formula depending on what they need to find.
Example:
An object experiences a force of 50 N and accelerates at . To find mass: kg.
A car has a mass of 1000 kg. You need to find what force is required to accelerate it at .
What force is needed?
Step 1: Write the mathematical expression
Use :
Finance: Simple Interest
Banks use $I = Prt$ to calculate interest. You can rearrange to find principal, rate, or time.
Example:
If you earned 60 dollars interest on an investment at 5% for 3 years, the principal was dollars.
You want to earn 100 dollars interest in 2 years with a principal of 1000 dollars.
What interest rate do you need?
Step 1: Write the mathematical expression
Solve for :
Geometry: Finding Dimensions
Construction workers often need to find a missing dimension when they know the area or perimeter.
Example:
A room has area 120 square feet and is 10 feet wide. The length is feet.
A rectangular garden has a perimeter of 56 meters. The length is 18 meters.
What is the width?
Step 1: Write the mathematical expression
Use , solve for :
Key Takeaways
- 1A literal equation contains two or more variables (like or )
- 2To solve for a variable, use inverse operations: undo addition with subtraction, undo multiplication with division
- 3Always apply the same operation to BOTH sides of the equation
- 4When dividing a sum, divide every term:
- 5Many real-world formulas (physics, finance, geometry) are literal equations
Frequently Asked Questions
Glossary
- Literal equation
- An equation containing two or more variables (letters), such as or
- Isolate
- To get a variable alone on one side of an equation
- Inverse operation
- An operation that undoes another (addition/subtraction, multiplication/division)
- Reciprocal
- The multiplicative inverse of a number; for , the reciprocal is
- Formula
- An equation that shows the relationship between different quantities
Formula Card
Area of Rectangle
Solve for $l$: $l = \frac{A}{w}$
Distance
Solve for $r$: $r = \frac{d}{t}$; for $t$: $t = \frac{d}{r}$
Perimeter of Rectangle
Solve for $l$: $l = \frac{P - 2w}{2}$
Simple Interest
Solve for $P$: $P = \frac{I}{rt}$
Celsius/Fahrenheit
Solve for $C$: $C = \frac{5(F-32)}{9}$