## Problem on the number of solutions to a linear system

How many solutions are there to $$\begin{pmatrix}1&1&1\\1&1&0\\0&0&1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}3\\2\\1\end{pmatrix}?$$ If there are any, find all of them.
• ## Solution

Observing that the linear system is of the form $\mathbf{A} x = b$ for a nonzero $b$, we note that this equation is inhomogeneous. Recall that for an inhomogeneous linear system
Hence, there are either no solutions or there are infinitely many.
We try to find a single solution to $$\begin{pmatrix}1&1&1\\1&1&0\\0&0&1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}3\\2\\1\end{pmatrix}$$ by writing out each of the three equations it corresponds to:
\begin{align} x + y + z &= 3\\ x + y \phantom{+ z} &= 2\\ \phantom{x + y + } z &=1 \end{align}
We plug the value of $z$ into the first equation to see \begin{align}x+y &=2\\x+y&=2 \end{align}
Hence a solution is any $(x,y,z)$ that satisfies $z=1$ and $x+y=2$. These are given by $$\begin{pmatrix}x\\2-x\\1\end{pmatrix}\text{ for any } x.$$