# SOLVING FOR A SPECIFIC VARIABLE

A formula is an equation that states the relationship between two or more variables. Formulas and some equations contain more than one variable. It is often useful to solve formulas for one of the variables.

Example 1 :

Solve the following equation for x.

3x - a = kx + b

Solution :

3x - a = kx + b

Subtract kx from each side.

3x - a - kx = b

3x - kx = b - a

Distributive property.

x(3 - k) = b - a

Divide each side by (3 - k).

x = b ⁻ ᵃ⁾⁄₍₃ ₋ k

Example 2 :

Solve the following equation for h.

A = ½ ⋅ (a + b)h

Solution :

A = ½ ⋅ (a + b)h

Multiply each side by 2.

2A = (a + b)h

Divide each side by (a + b).

²ᴬ⁄₍ₐ ₊ b = h

Example 3 :

Solve the following equation for h.

C = ⁵⁄₉ ⋅ (F - 32)

Solution :

C = ⁵⁄₉ ⋅ (F - 32)

Multiply each side by ⁹⁄₅

C⁄₅ = F - 32

C⁄₅ + 32 = F

Example 4 :

Solve the following equation for y.

2x + 3y = 18

Solution :

2x + 3y = 18

Subtract 2x from each side.

3y = 18 - 2x

Divide each side by 3.

y = ⁽¹⁸ ⁻ ²ˣ⁾⁄₃

y = ¹⁸⁄₃ - ²ˣ⁄₃

y = 6 - ²ˣ⁄₃

Example 5 :

Solve the following equation for x.

2x = ⁽ˣʸ ⁺ ³⁾⁄y

Solution :

2x = ⁽ˣʸ ⁺ ³⁾⁄y

Multiply each side by y.

2xy = xy + 3

Subtract xy from each side.

xy = 3

Divide each side by y.

x = ³⁄y

Example 6 :

Solve the following equation for y.

logx + logy = log(x + y)

Solution :

logx + logy = log(x + y)

Use fundamental law of logarithm on the left side of the equation.

log(xy) = log(x + y)

xy = x + y

Subtract y from each side.

xy - y = x

Use the Distributive property.

y(x - 1) = x

Divide each side by (x - 1).

y = ˣ⁄₍ₓ ₋ ₁₎

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