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 = sqrt ( 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 = sqrt ( 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 = + or - sqrt (H 2 - x 2)

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

    y = + sqrt (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 fomulas 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 Pi 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 fomulas 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 Pi h)


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


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

More references and links on how to Solve Equations, Systems of Equations and Inequalities.








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