**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)

and

E = mc^{2}

(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.

**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 = mc^{2}

**Solution : **

E = mc^{2}

Divide each side by m.

E / m = c^{2}

Take square root on each side.

√(E/m) = √c^{2}

√(E/m) = c

**Example 3 : **

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

S = 2∏r(h + r)

**Solution : **

S = 2∏r(h + r)

Use distributive property of multiplication over addition.

S = 2∏rh + 2∏r^{2}

Subtract 2∏r^{2 }from each side.

S - 2∏r^{2 } = 2∏rh

Divide each side by 2∏r.

(S - 2∏r^{2}) / 2∏r = h

**Example 4 : **

Solve for r in the formula for volume of sphere :

V = 4/3 ⋅ ∏r^{3}

**Solution : **

V = 4/3 ⋅ ∏r^{3}

Multiply each side by 3/4.

3V/4 = ∏r^{3}

Divide each side by ∏.

3V/4∏ = r^{3}

Take cube root on each side.

^{3}√(3V/4∏) = ^{3}√r^{3}

^{3}√(3V/4∏) = 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 of 3a + 5ac.

Q = a(3 + 5c)

Divide each side by (3 + 5c).

Q / (3 + 5c) = a

**Example 7 : **

Solve for b_{1} :

A = 1/2 ⋅ h(b_{1} + b_{2})

**Solution : **

A = 1/2 ⋅ h(b_{1} + b_{2})

Multiply each side by 2/h.

2A/h = b_{1} + b_{2}

Subtract b_{2} from each side.

2A/h - b_{2 }= b_{1}

**Example 8 : **

Solve for b :

a^{2} + b^{2} = c^{2}

**Solution : **

a^{2} + b^{2} = c^{2}

Subtract a^{2} from each side.

b^{2} = c^{2 }- a^{2}

Take square root on each side.

√b^{2} = √(c^{2 }- a^{2})

b = √(c^{2 }- a^{2})

After having gone through the stuff given above, we hope that the students would have understood, "Solving Literal Equations".

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