Math Problem Statement

Stan is a clerk at a 24-hour gas station. He gets paid $7.25 an hour to work during the day and $9.25 an hour to work at night, where x represents the amount of hours worked during the day, and y represents the amount of hours worked at night. Which graph represents the number of hours that he needs to work if Stan wants to make $300?

Solution

To solve this, let's break down the problem:

  1. Problem Analysis:

    • Stan earns $7.25 per hour during the day (represented by xx) and $9.25 per hour at night (represented by yy).
    • Stan wants to earn a total of $300.
  2. Setting Up the Equation: The total earnings equation is: 7.25x+9.25y=3007.25x + 9.25y = 300

  3. Finding the Graph Representation: We can rearrange the equation to find the intercepts:

    • For the xx-intercept (when y=0y = 0): 7.25x=300x=3007.2541.387.25x = 300 \Rightarrow x = \frac{300}{7.25} \approx 41.38
    • For the yy-intercept (when x=0x = 0): 9.25y=300y=3009.2532.439.25y = 300 \Rightarrow y = \frac{300}{9.25} \approx 32.43

    These intercepts give us the points (41.38,0)(41.38, 0) and (0,32.43)(0, 32.43), which should appear on the correct graph.

  4. Identifying the Correct Graph: Look for a graph with intercepts near x=41x = 41 and y=32y = 32, and that aligns with the negative slope expected from the equation.

The correct graph should be the one where:

  • The xx-axis intercept is approximately at 41.
  • The yy-axis intercept is approximately at 32.

Do you want further explanation on setting up and solving linear equations like this?

Related Questions

  1. How would the graph change if the total earnings target was $400 instead?
  2. What would the equation look like if Stan was paid $8 during the day and $10 at night?
  3. How many hours would Stan need to work only at night to earn $300?
  4. If Stan wanted to split his hours equally between day and night, how would his earnings equation change?
  5. How can we determine if a solution exists for a given earnings goal and hourly rates?

Tip: Always start by identifying intercepts when graphing linear equations with two variables—this simplifies plotting the line quickly.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Graphing

Formulas

7.25x + 9.25y = 300

Theorems

Linear Equation Intercepts

Suitable Grade Level

Grades 9-10