# 1.10 – Solving Literal Equations and Formulas

Objectives

• Solve literal equations with two variables.
• Rearrange formulas to solve for a quantity of interest in a real-world problem.
• Describe the steps in solving equations.
• Use units to guide and interpret your solutions of literal equations and formulas.

Key Terms

• Fahrenheit and Celsius – Ways to calculate and tell the temperature.
• In the USA, we use Fahrenheit. In Europe (and many other countries), they use Celsius.
• Formula for finding Celsius:
• $C=\frac{5}{9}(F-32)$
• What if you wanted to solve for Fahrenheit instead?
• Try solving for F by doing inverse (reverse) operations until you isolate the F.
• Step 1: Multiply by 9 on both sides of the equation
• $9C=9*\frac{5}{9}(F-32)$ and this will give you: $9C=5(F-32)$
• Step 2: Divide by 5 on both sides of the equation
• $\frac{9}{5}C=5\frac{(F-32)}{5}$ and this will give you: $\frac{9}{5}C=F-32$
• Step 3: Add 32 to both sides of the equation
• You get the formula for finding Fahrenheit: $\frac{9}{5}C+32=F$
• You can rewrite it as $F=\frac{9}{5}C+32$
• Formula – An equation that describes an important mathematical relationship.
• Literal Equation – An equation that involves two or more variables.
• Ohm’s Law  – Describes the relationships among voltage, current, and resistance in a conductor (ex. computer circuits)
• The voltage of the current is V.
• The amount of current flowing through the conductor is I
• Any resistance that is present is measured with R
• Formula: V = IR
• Example: If a voltage of 0.5 V (volts) is driving a current of 0.03 A (amps) through a resistor, what is the resistance fo the resistor, measured in ohms?
• The symbol for ohms is Omega: $\Omega$
• Step 1: List what we know and the formula
• V: 0.5
• I: 0.03
• Formula: V = IR
• Variable (unknown): R
• Step 2: Solve for R by dividing by I on both sides to isolate R
• $\frac{V}{I} = \frac{IR}{I}$
• $\frac{V}{I} = R$
• Step 3: Substitute and solve
• $\frac{0.5}{0.03} = R$
• $16.667 \Omega = R$
• The resistance of a computer circuit is 16.667 ohms
• Perimeter – The distance around an object
• Perimeter of a rectangle (formula): P = 2l + 2w
• P = perimeter
• l = length
• w = width
• Perimeter Example: Felicia has 60 feet of fencing. She’d like to use all of this fencing to enclose a rectangular space in her backyard for a vegetable garden. The length of the garden must be 10 feet. What will the width be?
• Step 1: List the facts & formula
• P = 2l + 2w
• Perimeter (fencing): 60ft
• Space: rectangular
• length: 10ft
• width: ?
• Step 2: Solve for w
• Step 3: Substitute known variables
• $\frac{60}{2} - 10 = w$
• Step 4: Simplify and solve
• $30 - 10 = w$
• w = 20
• The width of the garden is 20 feet long.

Notes

• Literal Equation Example 1: Solve for y
• Step 1: Write the problem
• $20x + 5y = 40$
• Step 2: Subtract 20x on both sides
• $5y = 40 - 20x$
• Step 3: Divide both sides by 5
• $\frac{5y}{5} = \frac{40 - 20x}{5}$
• Step 4: Separate the fraction on the right side into two terms
• $\frac{5y}{5} = \frac{40}{5} - \frac{20x}{5}$
• Step 5: Simplify
• $y = 8 - 4x$

• Literal Equation Example 2: Solve for x
• Step 1: Write the problem
• $20x + 5y = 40$
• Step 2: Subtract 5y on both sides
• $20x = 40 - 5y$
• Step 3: Divide both sides by 20
• $\frac{20x}{20} = \frac{40 - 5y}{20}$
• Step 4: Separate the fraction on the right side into two terms
• $\frac{20x}{20} = \frac{40}{20} - \frac{5y}{20}$
• Step 5: Simplify
• $x = 2 - \frac{y}{4}$