A cab company charges $2.75 per mile, plus an extra surcharge. Bernardo took a cab to the airport, which was 8 miles from his home. His total fare came to $29.50. Let $x$ represent the number of miles and $y$ represent the total cost of the cab ride.

AWhat is the slope-intercept equation of the line that fits this information?
BIf a customer paid $35, how many miles did they travel in the cab?

Question

03-11-2026
Mathematics

Answered

A cab company charges $2.75 per mile, plus an extra surcharge. Bernardo took a cab to the airport, which was 8 miles from his home. His total fare came to $29.50. Let $x$ represent the number of miles and $y$ represent the total cost of the cab ride.

AWhat is the slope-intercept equation of the line that fits this information?
BIf a customer paid $35, how many miles did they travel in the cab?

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T19:48:12-05:00

Determine the surcharge by substituting the known ride data into the linear equation $y = 2.75x + b$, resulting in $b = \$7.50$.
Define the complete slope-intercept equation for the cab fare as $y = 2.75x + 7.50$.
Solve for the distance traveled by setting the total cost to $\$35$ and calculating $x$ in the equation $35 = 2.75x + 7.50$.
Conclude that the equation is $y = 2.75x + 7.50$ and the distance traveled for $\$35$ is $\boxed{10 \text{ miles}}$.

Explanation

Determining the Slope-Intercept Equation

To find the slope-intercept equation of the total cost of a cab ride, we use the linear form $y = mx + b$, where $y$ represents the total cost, $x$ is the number of miles traveled, $m$ is the cost per mile, and $b$ is the fixed surcharge. We are given that the cost per mile $m = \$2.75$ and that an 8-mile ride costs $\$29.50$. We can substitute these values into the equation to determine the surcharge $b$:
\[29.50 = 2.75(8) + b\]
Calculating the product:
\[2.75 \times 8 = 22.00\]
Subtracting this from the total cost gives:
\[b = 29.50 - 22.00 = 7.50\]
Thus, the fixed surcharge is $\$7.50$, and the slope-intercept equation for the ride cost is:
\[y = 2.75x + 7.50\]

Calculating the Distance for a Given Fare

Now that we have the equation $y = 2.75x + 7.50$, we can determine the number of miles traveled for a total fare of $\$35$. We set $y = 35$ and solve for $x$:
\[35 = 2.75x + 7.50\]
First, subtract the surcharge from both sides:
\[35 - 7.50 = 2.75x\]
\[27.50 = 2.75x\]
Divide both sides by the cost per mile $2.75$:
\[x = \frac{27.50}{2.75} = 10\]
Therefore, a customer who paid $\$35$ traveled 10 miles. The slope-intercept equation is $y = 2.75x + 7.50$ and the distance for a $\$35$ fare is $10$ miles.

Examples

Understanding linear equations like $y = mx + b$ is essential in everyday financial management. For example, when choosing between different mobile phone plans with a base monthly fee and an additional per-minute cost, or calculating the cost of a rental service, you are essentially solving for a total price based on usage. This problem demonstrates how to model such real-world costs mathematically, helping you make informed decisions about your expenses.