Math Problem Statement

Tickets to the aquarium are $11 for adults and $6 for children. An after-school program has a budget of $200 for a trip to the aquarium. Which graph represents the cost constraint in this situation if the boundary line represents the equation 11x + 6y = 200?

Solution

To determine which graph correctly represents the cost constraint 11x+6y=20011x + 6y = 200, where xx is the number of adult tickets (costing $11 each) and yy is the number of child tickets (costing $6 each), we analyze the equation step by step.


Step 1: Rewrite the Equation in Slope-Intercept Form

We start with: 11x+6y=20011x + 6y = 200 To express yy in terms of xx, solve for yy: 6y=11x+2006y = -11x + 200 y=116x+2006y = -\frac{11}{6}x + \frac{200}{6} Simplify the y-intercept: y=116x+33.33y = -\frac{11}{6}x + 33.33

  • Slope = 116-\frac{11}{6} (negative slope, meaning the line decreases)
  • y-intercept = 33.3333.33 (approximately at 33 on the y-axis)

Step 2: Check Each Graph

  1. Graph A:

    • The line does not pass through (0,33.33)(0, 33.33); the intercept is incorrect.
    • Not correct.
  2. Graph B:

    • The line starts at (0,36)(0, 36) and has a steeper negative slope than 116-\frac{11}{6}.
    • Not correct.
  3. Graph C:

    • The line has a negative slope, and the intercept appears around y=33.33y = 33.33.
    • This aligns closely with the equation y=116x+33.33y = -\frac{11}{6}x + 33.33.
    • This graph looks correct.
  4. Graph D:

    • The line starts at (0,28)(0, 28), which is too low for the correct intercept.
    • Not correct.

Step 3: Final Answer

The graph that represents the equation 11x+6y=20011x + 6y = 200 is:

Graph C


Do you want a detailed explanation of how I verified the slope or intercept further? Let me know!


Related Questions:

  1. How do you graph a linear equation like 11x+6y=20011x + 6y = 200 step by step?
  2. What does the slope 116-\frac{11}{6} represent in this problem?
  3. How can you verify whether a point satisfies the equation 11x+6y=20011x + 6y = 200?
  4. How does the graph change if the budget increases to $300?
  5. How do you calculate the maximum number of child tickets if there are no adult tickets?

Tip:

To graph an equation efficiently, always solve for yy first and identify the slope and y-intercept. Then, test key points like intercepts to confirm accuracy.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form
Graphing Linear Equations

Formulas

Linear equation: y = mx + b
Cost constraint: 11x + 6y = 200

Theorems

Slope-Intercept Form Theorem

Suitable Grade Level

Grades 7-10