Solving Literal Equations – Tutorial

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 the literal equation:

\[ P = 2L + 2W \]

for \(W\).

Solution to Example 1

• 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: \[ W = \frac{P - 2L}{2} \]

Example 2

Solve the literal equation:

\[ H = \sqrt{x^2 + y^2} \]

for \(y\), given that \(H\), \(x\), and \(y\) are positive real numbers and \(H > x\) and \(H > y\).

Solution to Example 2

• 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 the equation:

\[ C = \frac{5}{9}(F - 32) \]

Solution to Example 3

• \[ C = \frac{5}{9}(F - 32) \]
• Multiply both sides by \(\frac{9}{5}\): \[ \frac{9}{5}C = F - 32 \]
• Add 32 to both sides: \[ F = \frac{9}{5}C + 32 \]

Example 4

Express \(y\) in terms of \(x\) for the literal equation:

\[ ax + by = c,\quad b \neq 0 \]

Solution to Example 4

• \[ ax + by = c \]
• Subtract \(ax\) from both sides: \[ by = -ax + c \]
• Divide by \(b\): \[ y = -\frac{a}{b}x + \frac{c}{b} \]

Exercises

Solve each literal equation for the indicated variable.

1. \(A = WL\), for \(L\)
2. \(y = mx + b\), for \(x\)
3. \(A = \frac{1}{2}(B + a)\), for \(a\)
4. \(S = 2\pi r h\), for \(r\)
5. \(F = \frac{9}{5}C + 32\), for \(C\)
6. \(\frac{1}{x} = \frac{1}{y} + \frac{1}{z}\), for \(y\)

Answers to Exercises

1. \(L = \frac{A}{W}\)
2. \(x = \frac{y - b}{m}\), with \(m \neq 0\)
3. \(a = 2A - B\)
4. \(r = \frac{S}{2\pi h}\)
5. \(C = \frac{5}{9}(F - 32)\)
6. \(y = \frac{xz}{z - x}\), with \(z \neq x\)

More References

Solve Equations, Systems of Equations, and Inequalities