Skill: Model real-world situations using linear equations

A linear equation shows how one quantity changes at a constant rate from a starting value.

In the form

$$y=mx+b$$

• $y$ is the total or changing quantity.
• $x$ is the input or variable.
• $m$ is the rate of change.
• $b$ is the fixed starting value.

Example

A taxi charges a fixed fee of $4 plus $2.50 for each mile traveled. How many miles can you travel for $19?

1. Choose the variables.

Let

• $x=$ number of miles traveled
• $y=$ total cost in dollars

2. Identify the fixed value and rate.

• The fixed fee is $4, so $b=4$.
• The cost increases by $2.50 for every mile, so $m=2.50$ dollars per mile.

3. Write the equation.

Using $y=mx+b$:

$$y=2.50x+4$$

This equation means the total cost is $2.50 for each mile, plus the $4 starting fee.

4. Use the given total cost.

The trip costs $19, so substitute $19$ for $y$:

$$19=2.50x+4$$

5. Solve for $x$.

Subtract $4$ from both sides:

$$15=2.50x$$

Divide by $2.50$:

$$x=6$$

6. Interpret the answer.

You can travel 6 miles for $19. The answer is meaningful because a distance cannot be negative, and $6$ miles fits the situation.