In an exercise about multiple regression, I'm given the following data matrix $X$ and error vector $\epsilon$:
$$ X = \begin{bmatrix} 1 & 4 & x \\ 1 & -2 & 1\\ 1 & 1 & 4 \\ 1 & -2 & 8 \\ 1 & y & 2 \\ \end{bmatrix}~, \quad \epsilon = \begin{bmatrix} z \\ 4 \\ 5 \\ -7 \\ -1 \\ \end{bmatrix} $$ I'm asked to find $x,y$ and $z$ but I don't know what to do. I thought the formula of $Y = \beta X + \epsilon$ might help but it doesn't do much since we don't know $\beta$ nor $Y$. Any help is appreciated.
I managed to find the answer: Through properties of the residuals we know that $X^{T}\epsilon = 0$ so all we need to do is solve the following system: \begin{align} z + 4 + 5 -7 - 1 = 0\\ 4z - 8 + 5 + 14 - y = 0\\ xz + 4 + 20 - 56 -2 = 0 \end{align} So that \begin{align} z = -1 \\ y = 7 \\ x = -34 \end{align}