<s>[INST] Not understanding how answer to vector question is arrived at I'm going through Multivariable and Vector Calculus by Sarhan Musa and David Santos, and I'm stuck on exercise 1.1.9.
The question says:

Let A,B be two points on the plane. Construct two points I and J such that $$\overline{IA}=-3\overline{IB}$$ $$\overline{JA} = -\frac{1}{3}\overline{JB}$$ and then demonstrate that for any arbitrary point M on the plane $$\overline{MA} + 3\overline{MB} = 4\overline{MI}$$ and $$3\overline{MA} + \overline{MB} = 4\overline{MJ}$$

I've drawn the question out and I can see vaguely geometrically how that could be the case. But I don't see how the author arrives at the answer (from the back of the book):

$$\begin{align}\overline{MA} + 3\overline{MB} &= 3\overline{MI}+\overline{MI}+\overline{IA}+3\overline{IB} \tag{1}\\&= 4\overline{MI} + \overline{IA} + 3\overline{IB}\tag{2}\\&= 4\overline{MI}\tag{3} \end{align}$$ and $$\begin{align}3\overline{MA} + \overline{MB} &= 3\overline{MJ} + 3\overline{JA}+\overline{MJ}+\overline{JB} \tag{4}\\ \tag{5}&=4\overline{MJ} + 3\overline{JA} + \overline{JB} \\ \tag{6}&= 4\overline{MJ}\end{align}$$

Specifically, how does the author arrive at the right hand side of lines 1 and 4?
 [/INST] For (1), we write:
\begin{align*}\overline{MA} + 3\overline{MB} 
&= \left(\overline{MI}+\overline{IA} \right) + 3\left(\overline{MI}+\overline{IB}\right)\\
&= \overline{MI}+\overline{IA} + 3\overline{MI} + 3\overline{IB}\\
&= 3\overline{MI} + \overline{MI}+\overline{IA} + 3\overline{IB}
\end{align*}
The same computation with $J$ instead of $I$ yields (4)
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