Q.Suppose $P_1$, $P_2$, $P_3$ and $R_1$, $R_2$, $R_3$ are as in Example 2. Let the firm have $330$ units of $R_1$, $455$ units of $R_2$ and $140$ units of $R_3$ available with it, and let the amount of raw materials $R_1$, $R_2$ and $R_3$ required to manufacture each unit of the three products be given by

[!FORMULA] $B = \begin{pmatrix} 3 & 4 & 0 \\ 7 & 9 & 3 \\ 5 & 12 & 7 \end{pmatrix}$

(rows $P_1, P_2, P_3$; columns $R_1, R_2, R_3$). How many units of each product are to be made so as to utilise the full available raw material?

Let $x, y, z$ be the units of $P_1, P_2, P_3$. Reading the columns of $B$ against the stock,

$3x + 7y + 5z = 330,\quad 4x + 9y + 12z = 455,\quad 3y + 7z = 140.$ …

Step 1 — Identify. Turn the "full utilisation" requirement into equations that pin down the production quantities.

Step 2 — Set up variables. Let the firm produce $x$ units of $P_1$, $y$ units of $P_2$ and $z$ units of $P_3$. Read $B$ down each raw-material column to see how much of that material all products together consume.

Step 3 — Mathematical formulation. Column $R_1$ of $B$ is $(3,7,5)$, so $R_1$ consumed is $3x+7y+5z$; setting it equal to the $330$ in stock, and doing the same for $R_2$ (column $(4,9,12)$) and $R_3$ (column $(0,3,7)$):

In matrix form,

Step 4 — Solve. Eliminate $x$: multiply the first equation by $4$ and the second by $3$ and subtract,

Substitute into the third equation $3y+7z=140$:

$3(16z-45) + 7z = 140 \ \Rightarrow\ 55z = 275 \ \Rightarrow\ z = 5,$ …

Method: Setting Up and Solving a Linear System from a Matrix "Recipe" Table

This method applies whenever a table (matrix) gives the amount of each resource consumed per unit of several products, and you must find production quantities that satisfy given resource totals.

Steps

Step 1: Assign unknowns to the quantities being found

Let a variable stand for the (unknown) number of units of each product to be made.

Step 2: Read the recipe matrix by columns, not rows

Since each row of the given matrix lists one product's consumption of every resource, the total consumption of one specific resource (summed across all products) is read down that resource's column — multiply each product's unknown quantity by its entry in that column and add.

Step 3: Set each resource's total consumption equal to the amount available

This produces one linear equation per resource, in the unknown production quantities — as many equations as there are resource constraints.

Step 4: Solve the resulting system of linear equations …

Common Mistakes

Mistake 1: Reading the matrix by rows instead of by columns

Why it's wrong: The given matrix $B$ has rows = products and columns = raw materials, so a single resource's total usage comes from that resource's column across all products — reading along a row instead mixes up different resources' consumption figures and produces a completely wrong equation. Correct approach: before writing any equation, explicitly state which axis of the matrix is products and which is raw materials, and read down the correct column.

Mistake 2: Arithmetic errors during elimination

Why it's wrong: Solving three simultaneous equations by elimination involves several multiplication and subtraction steps (e.g. multiplying one equation by 4 and another by 3 before subtracting); a slip in any of these steps propagates through the rest of the solution and can silently give an internally-consistent but wrong final answer. Correct approach: after solving, always substitute the found values back into all three original equations to check they hold exactly. …