Finding the equation of a line that is the intersection of two planes?

Question

Can someone please explain this part of the problem attached herewith? I have no idea what's going on in the highlighted region.

Thanks.

Answer 1

You are trying to eliminate fractions. Obviously $\lambda=8\mu+k$ since the fractions have a least common multiple of 8. Next, you want to figure out a possible value for $k$. So, multiplying out, the scalar part becomes:

$$\begin{bmatrix}\tfrac 7 4 \\ \tfrac 5 8 \\ 0\end{bmatrix} + k\begin{bmatrix}\tfrac 3 4 \\ \tfrac 1 8 \\ 1\end{bmatrix} = \begin{bmatrix}\tfrac{7+3k}{4} \\ \tfrac{5+k}{8} \\ k\end{bmatrix}$$

So we are looking for $7+3k\equiv 0 \pmod 4$ and $5+k\equiv 0 \pmod 8$. This gives $k=3$ as a possible value that will eliminate fractions, which is what was used.