Substitution Algebra: How to produce a "y equals" statement

Question

$x + 3y = -4$

$y + x = 0$

What are $x$ and $y$?

I know that in the first problem you replace the $y$ beside the $3$ but I cannot figure out how to turn $y + x = 0$ into a $y=$ statement

Answer 1

If you're supposed to use linear algebra and Gaussian elimination to solve this problem, then this is most likely what your thought process is supposed to be: $$\begin{bmatrix}1&3&-4\\1&1&0\end{bmatrix}\sim\begin{bmatrix}1&3&-4\\0&-2&4\end{bmatrix}\sim\begin{bmatrix}1&3&-4\\0&1&-2\end{bmatrix}\sim\begin{bmatrix}1&0&2\\0&1&-2\end{bmatrix}.$$ Thus, we can see that your system is consistent for the unique solution where $x=2$ and $y=-2$.

Alternatively: (linear algebra is kind of overkill) You can note that $x+y=0$ and thus $y=-x$. Substitute this into the equation $x+3y=-4$ and you will see that $$ x-3x=-4\Longleftrightarrow -2x=-4\Longleftrightarrow x=2. $$ Since $x=2$ and $y=-x$, then $y=-(2)=-2$. Done.