Algebra
Rearranging Equations
Solving for a Variable
Isolate a specific variable in an equation with multiple variables.
When solving for a specific variable in a multi-variable equation, treat all other variables as constants. The same inverse operation rules apply:
For , solve for :
- Subtract :
- Divide by :
The answer is expressed in terms of the remaining variables — this is perfectly valid in algebra.
Rearranging Formulas
Rewrite scientific and geometric formulas to solve for different quantities.
Formulas are literal equations that describe real-world relationships. Rearranging a formula lets you solve for any quantity it contains:
| Formula | Solve for | Result |
|---|---|---|
| (interest) | ||
| (area) | ||
| (temperature) |
The result is a reusable formula that works for any input values.
Literal Equations
Work with equations where coefficients are represented by letters.
A literal equation has coefficients represented by letters rather than specific numbers. The solving process is identical — just treat the letter coefficients as constants:
For , solve for :
- Subtract :
- Divide by :
This gives a general solution that works for any values of , , and (with ).
Concepts
What is a Literal Equation?
A literal equation is an equation that contains two or more variables. Instead of solving for a numerical value, you solve for one variable in terms of the others. For example, is a literal equation — you can solve for , , or in terms of the remaining variables.
The Same Rules Apply
Solving literal equations uses the same principles as solving numerical equations. The key difference is that your answer will contain other variables instead of numbers. Use inverse operations and maintain balance — treat the variable you are solving for as the "unknown" and everything else as constants.
Strategic Order of Operations
When isolating a variable, work in reverse order of operations (reverse PEMDAS):
- First, handle addition and subtraction (move terms without the target variable)
- Then handle multiplication and division
- Finally, deal with exponents, parentheses, and roots
This is sometimes called the "reverse PEMDAS" or "undoing" approach.
Keeping It General
The power of literal equations is generality. A formula like solved for gives — a reusable result that works for any values of . This is why formulas are preferred over numerical calculations in science.
Worked Examples
Solving for a Variable
Problem
Solution Steps
Answer
Rearranging a Formula
Problem
Solution Steps
Answer
Geometric Formula
Problem
Solution Steps
Answer