In the PME Journal volume 1, issue 9, archived in here, in page 359 it is given that
However, is there any justification for solving system of linear equation with this method ?
The author only provides some examples for this method in the continuation of the article, and there is not even a word why this method works.
They are (essentially) using a 'multiply and subtract' technique to simplify the equations.
E.g. Multiply 1) by $d_2$ and 2) by $d_1$. Subtract and you get 4).
You get 5) analogously with $d_3$ and $d_2$ on 2) and 3). Then, using the same idea, multiply 4) by $A_2$ and 5) by $A_1$, subtract and you get
$$|A_1B_2|y + |A_1C_2|z = 0$$
whence $\frac{y}{z} = \frac{-|A_1C_2|}{|A_1B_2|}$ or $y:z = -|A_1C_2|:|A_1B_2|$.
The same thing again gives the proportion for $x$. Once you have the proportions, you can substitute and solve for the values if necessary.