A literal equation is an equation that expresses a relationship between two or more variables. A formula is a common type of literal equation. This tutorial shows how to solve literal equations for a chosen variable, with step-by-step examples, solutions, and exercises.
How to Solve Literal Equations (Examples with Solutions)
Example 1: Solve \(P = 2L + 2W\) for \(W\)
- Given: \[ P = 2L + 2W \]
- Isolate the term containing \(W\): Add \(-2L\) to both sides: \[ P - 2L = 2L + 2W - 2L \]
- Simplify: \[ P - 2L = 2W \]
- Divide both sides by 2: \[ W = \dfrac{P - 2L}{2} \]
Example 2: Solve \(H = \sqrt{x^2 + y^2}\) for \(y\) (\(H, x, y > 0\), \(H > x\), \(H > y\))
- Given: \[ H = \sqrt{x^2 + y^2} \]
- Square both sides: \[ H^2 = x^2 + y^2 \]
- Subtract \(x^2\) from both sides: \[ H^2 - x^2 = y^2 \]
- Solve for \(y\): \[ y = \pm\sqrt{H^2 - x^2} \]
- Since \(y\) is positive: \[ y = \sqrt{H^2 - x^2} \]
Example 3: Express \(F\) in terms of \(C\) in \(C = \dfrac{5}{9}(F - 32)\)
- Given: \[ C = \dfrac{5}{9}(F - 32) \]
- Multiply both sides by \(\dfrac{9}{5}\): \[ \dfrac{9}{5}C = F - 32 \]
- Add 32 to both sides: \[ F = \dfrac{9}{5}C + 32 \]
Example 4: Express \(y\) in terms of \(x\) for \(ax + by = c\) (\(b \neq 0\))
- Given: \[ ax + by = c \]
- Subtract \(ax\) from both sides: \[ by = -ax + c \]
- Divide by \(b\): \[ y = -\dfrac{a}{b}x + \dfrac{c}{b} \]
Exercises and Solutions
Solve each literal equation for the indicated variable.
- \(A = WL\), for \(L\)
- \(y = mx + b\), for \(x\)
- \(A = \dfrac{1}{2}(B + a)\), for \(a\)
- \(S = 2\pi r h\), for \(r\)
- \(F = \dfrac{9}{5}C + 32\), for \(C\)
- \(\dfrac{1}{x} = \dfrac{1}{y} + \dfrac{1}{z}\), for \(y\)
Answers to Exercises
- \(L = \dfrac{A}{W}\)
- \(x = \dfrac{y - b}{m}\), with \(m \neq 0\)
- \(a = 2A - B\)
- \(r = \dfrac{S}{2\pi h}\)
- \(C = \dfrac{5}{9}(F - 32)\)
- \(y = \dfrac{xz}{z - x}\), with \(z \neq x\)