Literal Equations

What are Literal Equations?

Literal equation: Equation with multiple variables

Examples:

• A = lw (area of rectangle)
• C = 2πr (circumference)
• d = rt (distance formula)
• ax + by = c (standard form)

Goal: Solve for one variable in terms of the others

Why Solve Literal Equations?

Rearrange formulas to isolate the variable you need

Example: Distance formula d = rt

• Solve for r: r = d/t
• Solve for t: t = d/r

Use: Plug in known values after rearranging

Solving for a Variable

Use same algebra steps as solving regular equations:

1. Undo addition/subtraction
2. Undo multiplication/division
3. Apply inverse operations
4. Keep equation balanced

Example 1: Area of Rectangle

Solve for w: A = lw

Divide both sides by l:

A/l = lw/l
A/l = w

Answer: w = A/l

Example 2: Perimeter

Solve for b: P = 2a + 2b

Subtract 2a from both sides:

P - 2a = 2b

Divide by 2:

(P - 2a)/2 = b

Answer: b = (P - 2a)/2

Example 3: Distance Formula

Solve for t: d = rt

Divide by r:

d/r = t

Answer: t = d/r

Multi-Step Literal Equations

Combine multiple steps

Example 1: Slope-Intercept Form

Solve for x: y = mx + b

Subtract b:

y - b = mx

Divide by m:

(y - b)/m = x

Answer: x = (y - b)/m

Example 2: Standard Form

Solve for y: 3x + 2y = 12

Subtract 3x:

2y = 12 - 3x

Divide by 2:

y = (12 - 3x)/2

Or: y = 6 - (3/2)x

Answer: y = (12 - 3x)/2

Example 3: Area of Trapezoid

Solve for h: A = (1/2)(b₁ + b₂)h

Multiply by 2:

2A = (b₁ + b₂)h

Divide by (b₁ + b₂):

2A/(b₁ + b₂) = h

Answer: h = 2A/(b₁ + b₂)

Equations with Fractions

Clear fractions first by multiplying by LCD

Example 1: With Fraction

Solve for x: y/3 = x + 5

Multiply by 3:

y = 3(x + 5)
y = 3x + 15

Subtract 15:

y - 15 = 3x

Divide by 3:

(y - 15)/3 = x

Answer: x = (y - 15)/3

Example 2: Multiple Fractions

Solve for a: (a + b)/c = d

Multiply by c:

a + b = cd

Subtract b:

a = cd - b

Answer: a = cd - b

Distributing Before Solving

Use distributive property when needed

Example 1: Distributive Property

Solve for x: a(x + b) = c

Distribute a:

ax + ab = c

Subtract ab:

ax = c - ab

Divide by a:

x = (c - ab)/a

Answer: x = (c - ab)/a

Example 2: Complex Distribution

Solve for n: P = 2(l + w)

Distribute 2:

P = 2l + 2w

If solving for w:

P - 2l = 2w
(P - 2l)/2 = w

Answer: w = (P - 2l)/2 or w = P/2 - l

Temperature Conversions

Common literal equations

Example 1: Celsius to Fahrenheit

Given: F = (9/5)C + 32

Solve for C:

Subtract 32:

F - 32 = (9/5)C

Multiply by 5/9:

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

Answer: C = (5/9)(F - 32)

Example 2: Using the Formula

Convert 68°F to Celsius

Use C = (5/9)(F - 32):

C = (5/9)(68 - 32)
C = (5/9)(36)
C = 20°C

Interest Formulas

Simple interest: I = Prt

• I = interest
• P = principal
• r = rate (decimal)
• t = time (years)

Example: Solve for Principal

Solve for P: I = Prt

Divide by rt:

I/(rt) = P

Answer: P = I/(rt)

Use: If I = $150, r = 0.05, t = 2

P = 150/(0.05 × 2)
P = 150/0.1
P = $1500

Physics Formulas

Many physics equations use literal equations

Example 1: Density

Solve for m: D = m/V

Multiply by V:

DV = m

Answer: m = DV

Example 2: Kinetic Energy

Solve for v: KE = (1/2)mv²

Multiply by 2:

2·KE = mv²

Divide by m:

2·KE/m = v²

Square root:

√(2·KE/m) = v

Answer: v = √(2·KE/m)

Geometry Formulas

Common geometric formulas to rearrange

Example 1: Volume of Cylinder

Solve for h: V = πr²h

Divide by πr²:

V/(πr²) = h

Answer: h = V/(πr²)

Example 2: Surface Area of Rectangular Prism

Solve for w: SA = 2lw + 2lh + 2wh

Factor out terms with w:

SA = 2lw + 2wh + 2lh
SA - 2lh = 2lw + 2wh
SA - 2lh = w(2l + 2h)

Divide by (2l + 2h):

w = (SA - 2lh)/(2l + 2h)

Answer: w = (SA - 2lh)/(2l + 2h)

Strategy for Complex Equations

Step-by-step approach:

1. Identify the variable to solve for
2. Clear fractions (multiply by LCD)
3. Distribute if needed
4. Collect all terms with target variable on one side
5. Factor out target variable if necessary
6. Isolate target variable

Example: Complex Equation

Solve for x: ay - b = c(x + d)

Distribute c:

ay - b = cx + cd

Isolate term with x:

ay - b - cd = cx

Divide by c:

(ay - b - cd)/c = x

Answer: x = (ay - b - cd)/c

Real-World Applications

Engineering: Rearrange formulas for design specifications

Science: Solve for unknown variables in experiments

Business: Financial formulas (profit, cost, revenue)

Cooking: Scale recipes (solve for ingredient amounts)

Example: Scaling Recipe

Recipe for 4 servings uses 3 cups flour. How much for 10 servings?

Proportion: cups/servings = c/s

3/4 = x/10

Cross multiply:

4x = 30
x = 7.5 cups

Practice