The question is as follows.
A manufacturing process mixes three raw materials $M_1,$ $M_2,$ and $M_3$ to produce three products $P_1,$ $P_2,$ and $P_3.$ The ratios of the amounts of the raw materials (in the order $M_1,$ $M_2,$ $M_3$) which are used to make up each of the three products are as follows.
For $P_1$, the ratio is $1: 2: 3;$ for $P_2,$ the ratio is $1: 3: 5;$ and for $P_3,$ the ratio is $3: 5: 8.$
In a certain production run, 137 units of $M_1,$ 279 units of $M_2,$ and 448 units of $M_3$ were used. The problem is to determine how many units of each of the products $P_1,$ $P_2,$ and $P_3$ were produced in that run.
(a.) Set this problem up in matrix form. Use the letter $A$ for the matrix, and write down the (one-line) formula for the solution in matrix form.
(b.) Compute the inverse matrix of $A,$ and use it to solve for the production vector $P.$
(c.) Find a choice for the ratios for the third product (in lowest form) different from the other two ratios and for which the resulting system has non-unique solutions.
The solution for part (a.) states the equation for $P_1$ as $$P_1 = \frac{M_1} 6 + \frac{2M_2} 6 + \frac{3M_3} 6 = \frac{M_1} 6 + \frac{M_2} 3 + \frac{M_3} 2.$$ ($P_2$ and $P_3$ are found in a similar fashion.) However, my question is why wouldn't the equation be $$P_1 = M_1 + 2M_2 + 3M_3$$ (and similarly for $P_2$ and $P_3$)? There is nothing in the question that implies that 1 unit of $M_1$ (or $M_2$ or $M_3$) is the same amount of material as 1 unit of $P_1$ and thus that the sum of the coefficients must be one. Why can't 1 unit of $P_1$ simply consist of 1 unit of $M_1,$ 2 units of $M_2,$ and 3 units of $M_3?$ The ratio of its composition would remain exactly the same, and again, there's nothing I can parse from the question that implies that 1 unit of $M_1, M_2,$ or $M_3$ must be the same amount of material as 1 unit of $P_1, P_2,$ or $P_3.$
Considering that the ratio of $M_1$ to $M_2$ to $M_3$ for $P_1$ is $1 : 2 : 3,$ it follows that 1 unit of $P_1$ consists of 1 part $M_1,$ 2 parts $M_2,$ and 3 parts $M_3,$ hence there are a total of six parts of $P_1.$ Considering that $M_3$ makes up for three parts of $P_1,$ $M_2$ makes up for two parts of $P_1,$ and $M_1$ makes up for one part of $P_1,$ it follows that $$P_1 = \frac 1 6 M_1 + \frac 2 6 M_2 + \frac 3 6 M_3 = \frac 1 6 M_1 + \frac 1 3 M_2 + \frac 1 2 M_3.$$ Checking the units on the left- and right-hand side, if we set each of $P_1,$ $M_1,$ $M_2,$ and $M_3$ equal to $1,$ we obtain $1 = 1,$ hence the units are correct; however, if you use the equation $$P_1 = M_1 + 2 M_2 + 3 M_3,$$ then setting each of the variables to $1$ yields $1 = 6,$ and the units are not the same on both sides.