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.


Example 1:

Solve the formula
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 formula
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 formula
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 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 formulas 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.

Answers to Above Exercises:

Solve each of the formulas below for the indicated variable.
  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.

More References and links

Solve Equations, Systems of Equations and Inequalities.