Does a production mix of good A and B exist that uses up all resources

Question

Given that a firm produces goods $A$ and $B$, using inputs $F$ and $G$.Each unit of $A$ requires 8 units of $F$ and 10 units of $G$. Each unit of $B$ requires 4 units of $F$ and 15 units of $G$. There is a total of 240 units of $F$ and 600 units of $G$. Does a production mix of goods $A$ and $B$ exist which uses up all inputs $F$ and $G$.

My approach to this question was to model the two goods into the equations:
$A=8f+10g$
$B=4f+15g$

Where $f, g$ is the number of units of $F$ and $G$ used.
If we use up all the firms resources then the equation can be remodeled as:
$A=8f+10g$
$B=4(240-f)+15(600-g)$

But I'm not sure as to how to move forward with the question. Any help would be greatly appreciated.

Answer 1

I don't see that how can we add $f$ to $g$. I'll first draw a table summarising the given conditions.

\begin{array}{crrr} & A & B & \text{Total} \\ \hline F & 8 & 4 & 240 \\ G & 10 & 15 & 600 \end{array}

Let $a$ and $b$ be the amount of product $A$ and $B$ produced respectively. Then we have

$$\left\{ \begin{aligned} 8a + 4b &\le 240 \\ 10a + 15b &\le 600. \end{aligned} \right.$$

Since you're asking for the existence of "a production mix of goods $A$ and $B$ which uses up all inputs $F$ and $G$", the above inequalities are changed to equalities. This can be solved in many ways to get the solution $(a,b) = (15,30)$. For example, using inverse matrix: $\det{\begin{bmatrix}8 & 4 \\ 10 & 15\end{bmatrix}} = 80$.

\begin{align} \begin{bmatrix}8 & 4 \\ 10 & 15\end{bmatrix}^{-1} &= \frac{1}{80} \begin{bmatrix}15 & -4 \\ -10 & 8\end{bmatrix} = \begin{bmatrix}3/16 & -1/20 \\ -1/8 & 1/10\end{bmatrix} \\ \begin{bmatrix}x \\ y\end{bmatrix} &= \begin{bmatrix}3/16 & -1/20 \\ -1/8 & 1/10\end{bmatrix} \begin{bmatrix}240 \\ 600 \end{bmatrix} = \begin{bmatrix}15 \\ 30\end{bmatrix} \end{align}