# SOLVING LITERAL EQUATIONS

A literal equation is, simply put, an equation that has a lot of letters or variables.

For example,

A = π β w

(The formula for finding the area of a rectangle)

E = mc2

(Einsteinβs Theory of Relativity)

are both literal equations.

When a literal equation is given, we would often be asked to solve the equation for a given variable. The goal is to isolate that given variable. The process is the same process that we use to solve linear equations; the only difference is that we will be working with a lot more letters, and we may not be able to simplify as much as we can with linear equations.

## Solved Examples

Example 1 :

Solve for w in the formula for area of a rectangle :

A = π β w

Solution :

A = π β w

Divide each side by π.

α΄¬βπ = w

Example 2 :

Solve for c in the formula for Einsteinβs Theory of Relativity :

E = mc2

Solution :

E = mc2

Divide each side by m.

α΄±βm = c2

Take square root on each side.

β(α΄±βm) = βc2

β(α΄±βm) = c

Example 3 :

Solve for h in the formula for the surface area of a right cylinder :

S = 2Οr(h + r)

Solution :

S = 2Οr(h + r)

Use distributive property of multiplication over addition.

S = 2Οrh + 2Οr2

Subtract 2Οr2 from each side.

S - 2Οr= 2Οrh

Divide each side by 2Οr.

Example 4 :

Solve for r in the formula for volume of sphere :

V = β΄ββ β Οr3

Solution :

V = β΄ββ β Οr3

Multiply each side by 3/4.

Β³β±½ββ = Οr3

Divide each side by Ο.

Β³β±½ββΟ = r3

Take cube root on each side.

3β(Β³β±½ββΟ) = 3βr3

3β(Β³β±½ββΟ) = r

Example 5 :

Solve for w in the formula for perimeter of the rectangle :

P = 2(π + w)

Solution :

P = 2(π + w)

Use distributive property of multiplication over addition.

P = 2π + 2w

Subtract 2π from each side.

P - 2π = 2w

Divide each side by 2.

β½α΄Ύ β» Β²πβΎββ = w

Example 6 :

Solve for a :

Q = 3a + 5ac

Solution :

Q = 3a + 5ac

Factor a out from (3a + 5ac).

Q = a(3 + 5c)

Divide each side by (3 + 5c).

Qβββ β βcβ = a

Example 7 :

Solve for b1 :

A = Β½ β h(b1 + b2)

Solution :

A = Β½ β h(b1 + b2)

A = Κ°ββ β (b1 + b2)

Multiply each side by Β²βh.

Β²α΄¬βh = b1 + b2

Subtract b2 from each side.

Β²α΄¬βh - bb1

Example 8 :

Solve for b :

a2 + b2 = c2

Solution :

a2 + b2 = c2

Subtract a2 from each side.

b2 = c- a2

Take square root on each side.

βb2 = β(c- a2)

b = β(c- a2)

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