Lesson 5: Literal Equations and Real-World Applications

Property

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable.
Then, we begin to interpret the words as mathematical expressions using mathematical symbols.
For example, a variable cost can be written as $0.10x$, while a fixed cost is a constant added or subtracted, such as in $C = 0.10x + 50$.

To model a linear equation:

1. Identify known quantities.
2. Assign a variable to represent the unknown quantity.
3. If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
4. Write an equation interpreting the words as mathematical operations.
5. Solve the equation. Be sure the solution can be explained in words, including the units of measure.

Examples

• One number is 10 more than another, and their sum is 52. Let the first number be $x$. The second is $x+10$. The equation is $x + (x+10) = 52$, so $2x=42$, and $x=21$. The numbers are 21 and 31.
• A taxi charges 3 dollars plus 2 dollars per mile. The total cost $C$ for a ride of $x$ miles is modeled by the equation $C = 2x + 3$. A 5-mile ride costs $C = 2(5) + 3 = 13$ dollars.
• Two streaming services have different plans. Plan A is 15 dollars a month. Plan B is 5 dollars a month plus 2 dollars per movie. To find when they cost the same for $m$ movies, set $15 = 5 + 2m$. Solving gives $10 = 2m$, so $m=5$ movies.

Property

Literal equations are formulas that use letters to represent quantities and relationships. These equations can be rewritten to express one variable in terms of others. Using inverse operations, a chosen variable can be isolated on one side of the equation, revealing how it depends on the remaining quantities. The process is the same as solving a regular linear equation, but you will be working with variables instead of just numbers.

Examples

• The formula $d = rt$ represents the relationship between distance, rate, and time. If the distance is 120 miles and the time is 3 hours, substituting these values into the rewritten formula

gives

showing that the rate is 40 miles per hour.

• The formula $P = 2l + 2w$ represents the perimeter of a rectangle, where $l$ is the length and $w$ is the width. If the perimeter is 28 units and the length is 6 units, substituting these values into the rewritten formula

gives

showing that the width is 4 units.