# Find vector equation of a line given by a linear system

1) Write the vector equation of the line in $\mathbb R^3$ given by:

\begin{align} x+y+z&=6 \\ x+2y+z&=1 \end{align}

2) Write the vector equation of the plane in $\mathbb R^3$ that passing through (1,0,-1) and is perpendicular to the line of exercise 1)

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Subtracting the first equation from the second gives $y=-5$. In this plane, you want all the points satisfying $x+z=11$, which is a line.
Parametrically, we can give it as $(t,-5,11-t)$, for all real $t$.