# Solving Literal Equations - Tutorial

A literal equation is an equation that expresses a relationship between two or more variables. A formula is an example of a literal equation. We present a tutorial on how to solve literal equations for one of the variables. Detailed solutions to examples and answers to exercises are presented.

## How to Solve Literal Equations? Examples with Detailed Solutions

### Example 1

Solve the literal equation

P = 2L + 2W
for W.
### Solution to Example 1

- Given

P = 2L + 2W

- we first isolate the term containing W : add -2L to both sides of the equation

P - 2L = 2L + 2W - 2L

- Simplify to obtain

P - 2L = 2W

- Divide both sides by 2 to obtain W.

W = (P- 2L) / 2

### Example 2

Solve the literal equation
H = √ ( x^{ 2} + y^{ 2})

for y, where H, x and y are a positive real numbers and H is greater than x and greater than y.
### Solution to Example 2

- Given

H = √ ( x^{ 2} + y^{ 2})

- Square both sides

H^{ 2} = x^{ 2} + y^{ 2}

- Add - x
^{ 2} to both sides and simplify

H^{ 2} - x^{ 2} = x^{ 2} + y^{ 2} - x^{ 2}

H^{ 2} - x^{ 2} = y^{ 2}

- Solve for y taking the square root

y = ± √ (H^{ 2} - x^{ 2})

- Since y is a positive real number, then y is given by

y = + √ (H^{ 2} - x^{ 2})

### Example 3

Express F in terms of C in the literal equation

C = (5 / 9)(F - 32) .
### Solution to Example 3

C = (5 / 9)(F - 32)

- Multiply both sides of the formula by 9 / 5

(9 / 5) C = (9 / 5)(5 / 9)(F - 32)

- and simplify

(9 / 5) C = (F - 32)

- Add 32 to both sides of the formula.

(9 / 5) C + 32 = F

- The formula F = (9 / 5) C + 32 expresses F in terms of C.

### Example 4

Express y in terms of x in the literal equation

a x + b y = c , with b not equal to zero..
### Solution to Example 4

a x + b y = c

- Add - a x to both sides of the equation

a x + b y - a x = c - a x

b y = - a x + c

- Divide both sides by b.

y = - (a / b) x + c / b

### Exercises

Solve each of the literal equations below for the indicated variable.(see answers below).

- A = W L , for L.

- y = m x + b , for x.

- A = (1 / 2)(B + a) , for a.

- S = 2 π r h , for r.

- F = (9 / 5)C + 32 , for C.

- 1 / x = 1 / y + 1 / z , for y.

### Answers to Above Exercises:

- L = A / W

- x = (y - b) / m , for m not equal to zero.

- a = 2 A - B

- r = S / (2 π h)

- C = (5 / 9)(F - 32)

- y = (x z) / (z - x) , for z not equal to x.

__More References and links__

Solve Equations, Systems of Equations and Inequalities.