Simultaneous linear equations: EJC 2024 Promo Q1

Eunoia Junior College2024 PromoPromo Q1●●○ Standard4 marks

What this question tests

System of Linear Equations.

Question

A small furniture factory produces standardised stools, chairs and tables using both metal and plastic as raw materials. The table below shows the amount of each raw material required to make one unit of each type of furniture.

Furniture Item	Amount of raw material needed (in kg)
Furniture Item	Metal	Plastic
Stool	1	2
Chair	4	2
Table	10	4

Given $68$ kg of metal and $32$ kg of plastic, determine the number of each type of furniture that could be produced such that all three types of furniture are made, and all the raw materials are fully utilised. [4]

Show full worked solution▾

Let $S$ , $C$ and $T$ be the number of stools, chairs and tables produced respectively.

$\because$ $68$ kg of metal is used in all, $\begin{aligned} S + 4C + 10T &= 68. \end{aligned}$

$\because$ $32$ kg of plastic is used in all, $\begin{aligned} 2S + 2C + 4T &= 32. \end{aligned}$

Solving this system of linear equations produces $\begin{cases} S = -\dfrac{4}{3} + \dfrac{2}{3}T,\\[6pt] C = \dfrac{52}{3} - \dfrac{8}{3}T. \end{cases}$ Since $S \geq 1$ , $\begin{aligned} -\frac{4}{3} + \frac{2}{3}T &\geq 1\\ T &\geq 3.5. \end{aligned}$ Since $C \geq 1$ , $\begin{aligned} \frac{52}{3} - \frac{8}{3}T &\geq 1\\ T &\leq 6.125. \end{aligned}$ So $T=4,5$ or $6$ . Testing all cases,

When $T=4$ , $S$ is not integer.
When $T=5$ , $S=2$ , $C=4$ .
When $T=6$ , $S$ is not integer.

So the only solution is $S=2$ , $C=4$ , $T=5$ .

$\therefore$ The number of stools, chairs and tables that could be produced is $2$ , $4$ , and $5$ respectively.

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