Question

1. it costs $10 for marks meals-to-go business to make a chicken parmesan meal, and the business spends $2 per chicken parmesan meal in packaging. it costs the business $8 to make a fettuccini alfredo meal, and the business spends $1 per fettuccini alfredo meal in packaging. marks business can spend only $150 to make meals and $24 on packaging. identify the system of inequalities that represents the constraints for this scenario. let x = the number of chicken parmesan meals. let y = the number of fettuccini alfredo meals.

\\(\

$$\begin{cases}10x + 8y \\leq 150 \\\\ 2x + y \\leq 24 \\\\ x \\geq 0 \\\\ y \\geq 0\\end{cases}$$

\\)
\\(\

$$\begin{cases}10x + 8y \\leq 24 \\\\ x + y \\leq 150 \\\\ 2x \\geq 0 \\\\ y \\geq 0\\end{cases}$$

\\)
\\(\

$$\begin{cases}10x + 2y \\leq 150 \\\\ 8x + y \\leq 24 \\\\ x \\geq 0 \\\\ y \\geq 0\\end{cases}$$

\\)
\\(\

$$\begin{cases}10x + 8y \\leq 150 \\\\ 2x + y \\leq 24 \\\\ x \\leq 0 \\\\ y \\leq 0\\end{cases}$$

\\)

Explanation:

Step1: Analyze meal - making cost

The cost to make a chicken parmesan meal is $10, and for LXI0 chicken parmesan meals, the cost is LXI1. The cost to make a fettuccini alfredo meal is $8, and for \(y\) fettuccini alfredo meals, the cost is \(8y\). The total amount that can be spent on making meals is $150. So the inequality for the meal - making cost is \(10x + 8y\leq150\).

Step2: Analyze packaging cost

The packaging cost for a chicken parmesan meal is $2, so for LXI0 chicken parmesan meals, the packaging cost is LXI1. The packaging cost for a fettuccini alfredo meal is $1, so for \(y\) fettuccini alfredo meals, the packaging cost is \(y\). The total amount that can be spent on packaging is $24. So the inequality for the packaging cost is \(2x + y\leq24\).

Step3: Analyze non - negativity of the number of meals

The number of chicken parmesan meals \(x\) and the number of fettuccini alfredo meals \(y\) cannot be negative. So we have \(x\geq0\) and \(y\geq0\).

Now we check the options:

• Option 1: \(

$$\begin{cases}10x + 8y\leq150\\2x + y\leq24\\x\geq0\\y\geq0\end{cases}$$

\) matches our derived inequalities.

• Option 2: The inequalities for meal - making cost and packaging cost are reversed and other inequalities are incorrect.
• Option 3: The cost coefficients for the meals and packaging are mixed up.
• Option 4: The non - negativity inequalities are reversed ( \(x\leq0\) and \(y\leq0\) which is impossible as the number of meals can't be negative).

Answer:

\(

$$\begin{cases}10x + 8y\leq150\\2x + y\leq24\\x\geq0\\y\geq0\end{cases}$$

\) (the first option)