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Question #86858
The Make-It-Good manufacturing company produces three products, Widgets, Mingets, and Tringles.

During a given year they plan to produce a total of 13,000 units of these products.

The per unit production costs for Widgets, Mingets, and Tringles are $4,$5, and $7 respectively. The per unit profit for the Widgets, Mingets, and Tringles is$1, $2, and$3 respectively.

If the production costs are to be $70,000 and the desired profit is$27,000, how many of each product should the company produce?

Set up the solution: define variables and determine the equations. Then, solve the system of equations using any valid method, and answer the question asked in the problem.

Define variables and make the system of equations:

Let w - count of Widgets.

Let m - count of Mingets.

Let t - count of Tringes.

Then

4w - costs of Widgets,

5m - costs of Mingets,

7t - costs of Tringes.

And w - profit for Widgets

2m - profit for Mingets

3t - profit for Tringes.

@$\begin{cases} m + t + w = 13000 \\ 5m + 7t + 4w = 70000 \\ 2m + 3t + w = 2700 \end{cases}@$

Solve it:

@$\begin{pmatrix} 1 & 1 & 1 \\ 5 & 7 & 4 \\ 2 & 3 & 1 \end{pmatrix} \begin{pmatrix} 13000 \\ 70000 \\ 27000 \end{pmatrix}@$

@$\begin{pmatrix} 1 & 1 & 1 \\ 0 & 2 & -1 \\ 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} 13000 \\ 5000 \\ 1000 \end{pmatrix}@$

@$\begin{pmatrix} 1 & 3 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 0 \end{pmatrix} \begin{pmatrix} 18000 \\ 5000 \\ -4000 \end{pmatrix}@$@$\begin{pmatrix} 1 & 3 & 0 \\ 0 & 2 & -1 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 18000 \\ 5000 \\ 4000 \end{pmatrix}@$

@$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 6000 \\ -3000 \\ 4000 \end{pmatrix}@$

so m = 6000, w = 3000, t = 4000

Tringles - 3000 items

Widgets - 6000 items

Mingets - 4000 items

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