A literal equation is an equation that contains two or more variables, such as d = rt or A = lw. Instead of solving for a number right away, you rearrange the formula to isolate the variable you want. This skill matters because science, geometry, engineering, and finance all use formulas with many changing quantities.
Learning to rewrite formulas helps you use one relationship in many different situations.
To isolate a variable, treat the equation like a balance: whatever operation you do to one side, you must do to the other side. Work backward through the operations that are attached to the target variable, using inverse operations in a careful order. For example, from F = ma, dividing both sides by m gives a = F/m.
The goal is not to change the meaning of the formula, but to create an equivalent formula that makes the chosen variable easy to calculate.
Key Facts
- A literal equation is an equation with multiple variables, such as V = lwh.
- To isolate a variable, use inverse operations on both sides of the equation.
- If ax = b, then x = b/a, as long as a is not 0.
- If x/a = b, then x = ab, as long as a is not 0.
- For d = rt, solving for t gives t = d/r, as long as r is not 0.
- For A = 1/2 bh, solving for h gives h = 2A/b, as long as b is not 0.
Vocabulary
- Literal equation
- An equation that contains two or more variables and can be rearranged to solve for one of them.
- Formula
- A rule written as an equation that relates quantities in a consistent way.
- Isolate
- To get a chosen variable alone on one side of an equation.
- Inverse operation
- An operation that undoes another operation, such as addition undoing subtraction or division undoing multiplication.
- Equivalent equations
- Equations that have the same solutions because the same valid operation was applied to both sides.
Common Mistakes to Avoid
- Moving a term without doing the same operation to both sides is wrong because it breaks the balance of the equation.
- Dividing only one term in a sum is wrong because division must apply to the entire side or expression being divided.
- Forgetting parentheses when substituting expressions is wrong because it can change the order of operations and produce a different formula.
- Dividing by a variable without noting it cannot be zero is wrong because division by zero is undefined.
Practice Questions
- 1 Solve P = 2l + 2w for w. Then find w when P = 50 and l = 15.
- 2 Solve V = 1/3 Bh for h. Then find h when V = 120 and B = 24.
- 3 A student solves d = rt for r and writes r = t/d. Explain the correct rearrangement and why the student's answer does not match the original relationship.