Apolline is mowing lawns for a summer job. For every mowing job, she charges an initial fee plus a constant fee for each hour of work. Her fee for a $5$ -hour job, for instance, is $\$42$ . Her fee for a $3$ -hour job is $\$28$ . Let $y$ represent Apolline's fee (in dollars) for a single job that took $x$ hours for her to complete. Which of the following information about the graph of the relationship is given?

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One-step inequalities: word problems

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Q. Apolline is mowing lawns for a summer job. For every mowing job, she charges an initial fee plus a constant fee for each hour of work. Her fee for a

5

-hour job, for instance, is

\$42

. Her fee for a

3

-hour job is

\$28

. Let

y

represent Apolline's fee (in dollars) for a single job that took

x

hours for her to complete. Which of the following information about the graph of the relationship is given?

Identify Charges: Step $1$ : Identify the charges for different hours worked. Apolline charges $\$42$ for a $5$ -hour job and $\$28$ for a $3$ -hour job.
Calculate Hourly Rate: Step $2$ : Calculate the hourly rate. $\newline$ To find the hourly rate, we can use the equation for a line, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. Here, $y$ represents the total fee, and $x$ represents the hours worked. $\newline$ First, calculate the slope ( $m$ ) which is the rate per hour: $\newline$ $m = (\text{Change in } y) / (\text{Change in } x) = (\$42 - \$28) / (5 \text{ hours} - 3 \text{ hours}) = \$14 / 2 \text{ hours} = \$7$ per hour.
Calculate Initial Fee: Step $3$ : Calculate the initial fee (y-intercept, $b$ ). $\newline$ Using one of the points, say $(3, \$\$$ $28$ )\), and the slope from Step $2$ : $\newline$ $\$$ $28$ = $\$$ $7$ ( $3$ ) + $b$ $\newline$ $\$$ $28$ = $\$$ $21$ + $b$ $\newline$ $b$ = $\$$ $28$ - $\$$ $21$ = $\$$ $7$ .
Write Equation: Step $4$ : Write the equation of the line. $\newline$ The equation representing the relationship between hours worked $x$ and the fee charged $y$ is: $\newline$ $y = 7x + 7$ .
Determine Graph Information: Step $5$ : Determine the information about the graph. $\newline$ The equation $y = 7x + 7$ is a linear equation, which means the graph is a straight line. The slope of the line is $7$ , indicating that the fee increases by $\$7$ for each additional hour worked. The y-intercept is $\$7$ , meaning the initial fee before any hours are worked is $\$7$ .