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 SolutionsExample 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).
 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 linksSolve Equations, Systems of Equations and Inequalities.
