# SOLVING LITERAL EQUATIONS

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

For example,

A = 𝓁 ⋅ w

(The formula for finding the area of a rectangle)

E = mc2

(Einstein’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 = 𝓁 ⋅ w

Solution :

A = 𝓁 ⋅ w

Divide each side by 𝓁.

ᴬ⁄𝓁 = w

Example 2 :

Solve for c in the formula for Einstein’s Theory of Relativity :

E = mc2

Solution :

E = mc2

Divide each side by m.

ᴱ⁄m = c2

Take square root on each side.

√(ᴱ⁄m) = √c2

√(ᴱ⁄m) = c

Example 3 :

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

S = 2πr(h + r)

Solution :

S = 2πr(h + r)

Use distributive property of multiplication over addition.

S = 2πrh + 2πr2

Subtract 2πr2 from each side.

S - 2πr= 2πrh

Divide each side by 2πr.

Example 4 :

Solve for r in the formula for volume of sphere :

V = ⁴⁄₃ ⋅ πr3

Solution :

V = ⁴⁄₃ ⋅ πr3

Multiply each side by 3/4.

³ⱽ⁄₄ = πr3

Divide each side by π.

³ⱽ⁄₄π = r3

Take cube root on each side.

3√(³ⱽ⁄₄π) = 3√r3

3√(³ⱽ⁄₄π) = r

Example 5 :

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

P = 2(𝓁 + w)

Solution :

P = 2(𝓁 + w)

Use distributive property of multiplication over addition.

P = 2𝓁 + 2w

Subtract 2𝓁 from each side.

P - 2𝓁 = 2w

Divide each side by 2.

⁽ᴾ ⁻ ²𝓁⁾⁄₂ = 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).

Q⁄₍₃ ₊ ₅c₎ = a

Example 7 :

Solve for b1 :

A = ½ ⋅ h(b1 + b2)

Solution :

A = ½ ⋅ h(b1 + b2)

A = ʰ⁄₂ (b1 + b2)

Multiply each side by ²⁄h.

²ᴬ⁄h = b1 + b2

Subtract b2 from each side.

²ᴬ⁄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.

b2 = √(c- a2)

b = √(c- a2)

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