For
$$x − y + 2z = −2$$ $$2x + 3y + 4z = 7$$ $$4x − 7y + 5z = k$$ $$8x − 4y + 6z = 2$$
Using Gaussian Elimination, I first get the below by applying $R2 - 2R1$, $R3 - 4R1$, $R4 - 8R1$
\begin{bmatrix} 1&-1&2&-2\\ 0&5&0&11\\ 0&-3&-3&k+8\\ 0&4&-10&18\\ \end{bmatrix}
Then $R3 + \frac{3}{5}R2$ and $R4 - \frac{4}{5}R2$
\begin{bmatrix} 1&-1&2&-2\\ 0&5&0&11\\ 0&0&-3&k+\frac{73}{5}\\ 0&0&-10&\frac{46}{5}\\ \end{bmatrix}
Finally, $R4 -\frac{10}{3}R3$
\begin{bmatrix} 1&-1&2&-2\\ 0&5&0&11\\ 0&0&-3&k+\frac{73}{5}\\ 0&0&0&\frac{-10k}{3} - \frac{-146}{3}\\ \end{bmatrix}
So for the system to be consistent, $k = -14.6$?
I have performed the calculations a number of times and came to different values of $k$ but this one appeared the most times so I think it is the correct answer?
To make it a bit simpler, consider equations
$$x − y + 2z = −2$$ $$2x + 3y + 4z = 7$$ $$8x − 4y + 6z = 2$$
Solve them for $x, y, z$ and plug the computed values into $$4x − 7y + 5z$$. If the initial system is to be consistent, $k$ must equal the computed value.