Consider three planes given by the Cartesian equations
![\[ 3x + y + z = 5, \qquad 3x + y + 5z = 7, \qquad x - y + 3z = 3. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-f8f917fc9f8e3c7dfce3d0dc55c45584_l3.png "Rendered by QuickLaTeX.com")
Find all of the points which are on the intersections of these three planes.
The points on the intersection must satisfy the system of equations
To find the points satisfying these equations we row-reduce the coefficient matrix of the system (if you don’t have any linear algebra, then you can use Gaussian elimination the long way),
![\begin{align*} \begin{pmatrix*}[r] 3 & 1 & 1 & 5 \\ 3 & 1 & 5 & 7 \\ 1 & -1 & 3 & 3 \end{pmatrix*} &\rightarrow \begin{pmatrix*}[r] 0 & 4 & -8 & -4 \\ 0 & 4 & -4 & -2 \\ 1 & -1 & 3 & 3 \end{pmatrix*} \\[9pt] &\rightarrow \begin{pmatrix*}[r] 1 & -1 & 3 & 3 \\ 0 & 1 & -2 & -1 \\ 0 & 2 & -2 & -1 \end{pmatrix*} \\[9pt] &\rightarrow \begin{pmatrix*}[r] 1 & 0 & 1 & 2 \\ 0 & 1 & -2 & -1 \\ 0 & 0 & 2 & 1 \end{pmatrix*} \\[9pt] & \rightarrow \begin{pmatrix*}[r] 1 & 0 & 0 & \frac{3}{2} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2} \end{pmatrix*}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ef15fb452f40dbcb072d70822c00d75b_l3.png "Rendered by QuickLaTeX.com")
Therefore, we have 
, 
, and 
. So, the only point on the intersection is the point 
.