Is there a better/easier way to solve this matrix?

Question

\begin{equation*} \begin{bmatrix} 4 & -1 & -1 & 0 &|&30 \\ -1 & 4 & 0 & -1&|&60 \\ -1 & 0 & 4 & -1&|&40 \\ 0 & -1 & -1 & 4&|&70 \end{bmatrix} \end{equation*}

What's the best way to solve the matrix above? There's a clear pattern of the diagonal 4's and 0's and the -1's so I feel like there has to be a better way of doing things rather than using scaling and row reduction.

If I do those methods I end up with messy fractions.

My Step 1:

New Row 2 = (1/4)Row 1 + Row 2

Even at step 1 I can tell the whole thing will be messy with fractions.

Is there a better way to solve this matrix? Or am I doing it wrong? Thanks.

Answer 1

Add all four rows to get $$\begin{pmatrix}2 &2 &2 &2\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}= 200 $$

or

$$\begin{pmatrix}1 &1 &1 &1\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}= 100 $$

Add that to every row to get $$\begin{pmatrix}5 &0 &0 &1\\ 0 & 5 & 1 & 0\\ 0 & 1 & 5 & 0\\ 1 & 0 & 0 & 5\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}= \begin{pmatrix}130\\160\\140\\170\end{pmatrix} $$

Now you have two separated 2x2 systems. Can you take it from here?