Question 532·Medium·Algebra

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The table shows a linear relationship between the number of miles driven, $m$ , by a taxi and the total cost, $C$ , in dollars.

Miles driven	Total cost (dollars)
6	15.75
9	20.25
12	24.75

Which equation represents the linear relationship between $m$ and $C$ ?

A table of a linear relationship gives mileage-cost pairs, and the correct equation must work for every pair. For multiple-choice equations, put one row into each distinct left side using Desmos, then compare the results with the constants on the right. One row is enough only if it separates every choice. If choices still match, check another row rather than trusting a single matching point.

Hints

Hint 1
Each row is a solution pair: a mileage and the cost for that same trip. Put both numbers from one row into an equation. Which different left sides do the options ask you to evaluate?
Hint 2
Test the same row in each left side, then compare each result with the number on the right. The sign matters: a result of $13.5$ does not equal $-13.5$ .

Step-by-step

Approach 1: Test one table row
Step 1
Evaluate the first left side
A solution pair is a mileage and cost that make the equation true. Use the first row, $m=6$ and $C=15.75$ . Type $2(15.75)-3(6)$ in Desmos; it prints $13.5$ . So an equation with the left side $2C-3m$ needs the positive $13.5$ , not $-13.5$ .
Step 2
Rule out the other left side
Keep the same row and type $3(15.75)-2(6)$ . Desmos prints $35.25$ , which is neither $13.5$ nor $-13.5$ . So both equations with $3C-2m$ fail. The equation that fits this taxi-cost row is $2C-3m=13.5$ . Choice D.

Approach 2: Build the cost rule
Step 1
Find the rate and starting cost
Enter miles as $x_1$ and cost as $y_1$ in a Desmos table, then type $y_1\sim ax_1+b$ . A linear regression fits a line to those pairs. Under PARAMETERS, Desmos shows $a=1.5$ and $b=6.75$ ; all three rows lie on that line.
Step 2
Write the taxi-cost equation
In the table, $x_1$ means miles $m$ and $y_1$ means cost $C$ , so the fitted line is $C=1.5m+6.75$ . The slope $1.5$ is dollars per mile; $6.75$ is the zero-mile cost. Don't use $15.75$ as that starting cost: its row has $6$ miles, not $0$ .
Step 3
Match the coefficient of cost
Multiply every term of $C=1.5m+6.75$ by $2$ to match the $2C$ in the choices:
$2C=3m+13.5.$
Step 4
Put the equation in the choices' form
Subtract $3m$ from both sides: $2C-3m=13.5$ . The positive constant comes from twice the zero-mile cost, so this equation represents the taxi's cost. Choice D.

Question 532·Medium·Algebra

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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