# 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 = ~+mn~ √ (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).
1. A = W L , for L.

2. y = m x + b , for x.

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

4. S = 2 π r h , for r.

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

6. 1 / x = 1 / y + 1 / z , for y.

1. L = A / W

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

3. a = 2 A - B

4. r = S / (2 π h)

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

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