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Michael has [tex]$\$[/tex]15[tex]$ and wants to buy a combination of cupcakes and fudge to feed at least three siblings. A cupcake costs $[/tex]\[tex]$2$[/tex], and a piece of fudge costs [tex]$\$[/tex]3[tex]$. This system of inequalities models the scenario:

\[
\begin{array}{l}
2x + 3y \leq 15 \\
x + y \geq 3
\end{array}
\]

\ \textless \ strong\ \textgreater \ Part A:\ \textless \ /strong\ \textgreater \ Describe the graph of the system of inequalities, including shading and the types of lines graphed. Provide a description of the solution set. (4 points)

\ \textless \ strong\ \textgreater \ Part B:\ \textless \ /strong\ \textgreater \ Is the point $[/tex](5,1)$ included in the solution area for the system? Justify your answer mathematically. (3 points)

Part C: Choose a different point in the solution set and interpret what it means in terms of the real-world context. (3 points)



Answer :

GinnyAnswer
Certainly! Let's go through each part of the problem step-by-step.

### Part A: Description of the Graph

We have the following system of inequalities:
[tex]\[ \begin{array}{l} 2x + 3y \leq 15 \\ x + y \geq 3 \end{array} \][/tex]

1. Graphing the Inequalities:

- For the inequality [tex]\(2x + 3y \leq 15\)[/tex]:
- The boundary line is [tex]\(2x + 3y = 15\)[/tex].
- To graph this, find the intercepts:
- When [tex]\(x = 0\)[/tex]: [tex]\(3y = 15 \Rightarrow y = 5\)[/tex]. This gives the point [tex]\((0, 5)\)[/tex].
- When [tex]\(y = 0\)[/tex]: [tex]\(2x = 15 \Rightarrow x = 7.5\)[/tex]. This gives the point [tex]\((7.5, 0)\)[/tex].
- Plot these points and draw the line. Since the inequality is [tex]\(\leq\)[/tex], shade the area below and including the line.

- For the inequality [tex]\(x + y \geq 3\)[/tex]:
- The boundary line is [tex]\(x + y = 3\)[/tex].
- To graph this, find the intercepts:
- When [tex]\(x = 0\)[/tex]: [tex]\(y = 3\)[/tex]. This gives the point [tex]\((0, 3)\)[/tex].
- When [tex]\(y = 0\)[/tex]: [tex]\(x = 3\)[/tex]. This gives the point [tex]\((3, 0)\)[/tex].
- Plot these points and draw the line. Since the inequality is [tex]\(\geq\)[/tex], shade the area above and including the line.

2. Solution Set:
- The solution set is the region where the shadings of the two inequalities overlap.
- The lines are solid because the inequalities are [tex]\(\leq\)[/tex] and [tex]\(\geq\)[/tex], which include the boundaries.

### Part B: Checking the Point (5,1)

Given the point [tex]\((5,1)\)[/tex]:

1. Check the inequality [tex]\(2x + 3y \leq 15\)[/tex]:
[tex]\[ 2(5) + 3(1) = 10 + 3 = 13 \leq 15 \][/tex]
This condition is satisfied.

2. Check the inequality [tex]\(x + y \geq 3\)[/tex]:
[tex]\[ 5 + 1 = 6 \geq 3 \][/tex]
This condition is also satisfied.

Since both conditions are satisfied, the point [tex]\((5,1)\)[/tex] is included in the solution area.

### Part C: Choosing a Different Point

Let's choose the point [tex]\((3,1)\)[/tex]:

1. Check the inequality [tex]\(2x + 3y \leq 15\)[/tex]:
[tex]\[ 2(3) + 3(1) = 6 + 3 = 9 \leq 15 \][/tex]
This condition is satisfied.

2. Check the inequality [tex]\(x + y \geq 3\)[/tex]:
[tex]\[ 3 + 1 = 4 \geq 3 \][/tex]
This condition is also satisfied.

Since both conditions are satisfied, the point [tex]\((3,1)\)[/tex] is in the solution set.

Interpretation:
- If Michael buys 3 cupcakes and 1 piece of fudge:
- The total cost would be [tex]\(2 \times 3 + 3 \times 1 = 6 + 3 = 9\)[/tex] dollars, which is within his budget of \$15.
- This combination would feed [tex]\(3 + 1 = 4\)[/tex] siblings, which meets the requirement of feeding at least 3 siblings.

Thus, the chosen point [tex]\((3,1)\)[/tex] provides a feasible and valid combination of cupcakes and fudge within the given constraints.

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