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For the following system, give a succinct description of the set of solutions.

3x - 4y + z = 12

Solution in the textbook (the matrices should be placed side by side):

\begin{bmatrix}x\\y\\z\end{bmatrix} \begin{bmatrix}4\\0\\0\end{bmatrix} \begin{bmatrix}4/3\\1\\0\end{bmatrix} \begin{bmatrix}-1/3\\0\\1\end{bmatrix}

I understand how they got the values for the rows corresponding to x, but I don't understand where the values for y and z came from? (As in I don't understand the 2nd and 3rd row for each tuple)

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The solution in the textbook has to be interpreted as:

$$s:= \begin{bmatrix}4\\0\\0\end{bmatrix}$$ $$h_1:=\begin{bmatrix}4/3\\1\\0\end{bmatrix}$$ $$h_2:=\begin{bmatrix}-1/3\\0\\1\end{bmatrix}$$

and all solutions of your equation can be expressed (where $\lambda_1, \lambda_2\in \mathbb{R}$) as

$$s+ \lambda_1 h_1 +\lambda_2 h_2 $$

Note that $h_1, h_2$ span the the solution space of the homogenuous problem $$3x - 4y + z = 0$$ and $s$ is one solution of the special problem $$3x - 4y + z = 12$$. Remember that each solution of the special problem is one solution of the special problem plus an element of solution space of the homogenuous problem.

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