Defining Linear Systems from Ratio (word question from MIT multivarible calculus pset2) source:https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-b-matrices-and-systems-of-equations/problem-set-2/MIT18_02SC_pset2.pdf

I am working through problem set 2 of 18.02sc on MIT OCW, Multivariable calculus. I don't understand the answer given for problem 2 part A, where you have to solve the linear equation. I understand the explanation but don't get the jump from their to the answer given. Many thanks for any help. Pics are below, of q and a.

The part I don't understand is part A) getting from the word question to the linear system (and is A°p =m), clearly A is in fact the ratios and P and m are just p1, p2, p3 and m1 etc but Im not clear on how they got A setup like they did with ratios as columns ie. Col 1 is the ratios of m1,m2,m3 for p1, col2 is for p2 etc, I would've thought it was A°m=P with the ratios as rows for each P it row 1 is p1?

A manufacturing process mixes three raw materials M1, M2, and M3 to produce three
products P1, P2, and P3. The ratios of the amounts of the raw materials (in the order
M1, M2, M3) which are used to make up each of the three products are as follows:
For P1 the ratio is 1: 2: 3; for P2 the ratio is 1: 3: 5; and for P3 the ratio is 3: 5: 8.
In a certain production run, 137 units of M1, 279 units of M2, and 448 units of M3
were used. The problem is to determine how many units of each of the products P1,
P2, and P3 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.

ANSWER
First you have to notice that the restrictions are raw material restrictions.

If you produce the amount $P_1$ you need $\frac16 P_1$ of raw material $M_1$.

If you produce the amount $P_2$ you need $\frac19 P_2$ of raw material $M_1$.

If you produce the amount $P_3$ you need $\frac3{16} P_3$ of raw material $M_1$.

Is this comprehensible ?

Thus the restriction for raw material 1 is

$\frac16 P_1+\frac19 P_2+\frac3{16} P_3=137$

Similar for the restrictions for $M_2$ and $M_3$

$\frac26 P_1+\frac39 P_2+\frac5{16} P_3=279$

$\frac36 P_1+\frac59 P_2+\frac8{16} P_3=448$

The Matrix equation  is $A\cdot \mathbf P=\mathbf m$

In your case

$\begin{pmatrix} \frac16&amp; \frac19 &amp;\frac3{16} \\ \frac26 &amp;\frac39 &amp;\frac5{16}  \\ \frac36 &amp;\frac59 &amp;\frac8{16}\end{pmatrix}\cdot \begin{pmatrix} P_1 \\ P_2 \\ P_3\end{pmatrix}=\begin{pmatrix} 137 \\ 279 \\ 448\end{pmatrix}$

If you calculate the LHS you will see that you will get the three equations above. The kind of the equation comes from the rules of matrix calculation. 

You  would get the equations as well if you would transpose both sides:

$(A\cdot \mathbf P)^T=\mathbf m^T$

$\mathbf P^T \cdot A^T=m^T$

$\begin{pmatrix} P_1 &amp; P_2 &amp; P_3\end{pmatrix}\cdot A^T=\begin{pmatrix} 137 &amp; 279 &amp; 448\end{pmatrix}$

Check both equations.
First you have to notice that the restrictions are raw material restrictions.

If you produce the amount $P_1$ you need $\frac16 P_1$ of raw material $M_1$.

If you produce the amount $P_2$ you need $\frac19 P_2$ of raw material $M_1$.

If you produce the amount $P_3$ you need $\frac3{16} P_3$ of raw material $M_1$.

Is this comprehensible ?

Thus the restriction for raw material 1 is

$\frac16 P_1+\frac19 P_2+\frac3{16} P_3=137$

Similar for the restrictions for $M_2$ and $M_3$

$\frac26 P_1+\frac39 P_2+\frac5{16} P_3=279$

$\frac36 P_1+\frac59 P_2+\frac8{16} P_3=448$

The Matrix equation  is $A\cdot \mathbf P=\mathbf m$

In your case

$\begin{pmatrix} \frac16&amp; \frac19 &amp;\frac3{16} \\ \frac26 &amp;\frac39 &amp;\frac5{16}  \\ \frac36 &amp;\frac59 &amp;\frac8{16}\end{pmatrix}\cdot \begin{pmatrix} P_1 \\ P_2 \\ P_3\end{pmatrix}=\begin{pmatrix} 137 \\ 279 \\ 448\end{pmatrix}$

If you calculate the LHS you will see that you will get the three equations above. The kind of the equation comes from the rules of matrix calculation. 

You  would get the equations as well if you would transpose both sides:

$(A\cdot \mathbf P)^T=\mathbf m^T$

$\mathbf P^T \cdot A^T=m^T$

$\begin{pmatrix} P_1 &amp; P_2 &amp; P_3\end{pmatrix}\cdot A^T=\begin{pmatrix} 137 &amp; 279 &amp; 448\end{pmatrix}$

Check both equations.