What is the line described by $x + y = y + z = \frac{x+z}{2}$ I'm asked to find the line in $R^3$ representated by $x + y = y + z = \frac{x+z}{2}$

I can see that $$x = z$$ $$y = 0$$

And I know that a line in $\mathbb{R}^3$ can be represented by $$ax+by+c=0$$

How do I proceed?
There may be faster approaches based on intuition, but I find the following approach based on solving a linear system of equation to be quite systematic.

Firstly, write it in matrix-vector form:
\begin{equation}
\left(\begin{array}{rrr}
1&amp;0&amp;-1\\
0&amp;1&amp;0
\end{array}\right)
\left(\begin{array}{c}
x\\
y\\
z
\end{array}\right)
=\left(\begin{array}{c}
0\\
0\\
0
\end{array}\right) \quad .
\end{equation}

Secondly, derive the extended coefficient matrix from it. Make sure to add a third line, so there is one line for each variable:
\begin{equation}
\left(\begin{array}{rrr|r}
1 &amp; 0 &amp; -1 &amp; 0\\
0 &amp; 1 &amp;  0 &amp; 0\\
0 &amp; 0 &amp;  0 &amp; 0\\
\end{array}\right) \quad .
\end{equation}

Thirdly, solve the underdetermined system of equations by introducing a parameter $\lambda$:
\begin{equation}
\left(\begin{array}{rrr|r}
1 &amp; 0 &amp; -1 &amp; 0\\
0 &amp; 1 &amp;  0 &amp; 0\\
0 &amp; 0 &amp;  1 &amp; \lambda\\
\end{array}\right) \rightarrow
\left(\begin{array}{rrr|r}
1 &amp; 0 &amp;  0 &amp; \lambda\\
0 &amp; 1 &amp;  0 &amp; 0\\
0 &amp; 0 &amp;  1 &amp; \lambda\\
\end{array}\right) \quad .
\end{equation}

Et voilà, you have a parametric representation of your line:
\begin{equation}
\vec{x} = \lambda\left(\begin{array}{c}
1\\
0\\
1
\end{array}\right) \quad .
\end{equation}
From x+y=y+z
It can be deduced that x=z
And from x+y = 0.5(x+z) while x= z
x + y = 0.5(x+x) = x

Thus y = 0

thus the equation can be concluded that
X=Z
Which is the equation of the line