1.x Intersections

HOW TO FIND

EXAMPLE

• find the intersection of $x+3z=5$ and $x+y+z=7$ as a vector equation

$$\begin{array}{r}x+3z=5\\ x+y+z=7\end{array}⟹\left[\begin{array}{cccc}1& 0& 3& 5\\ 1& 1& 1& 7\end{array}\right]RREF⟹\left[\begin{array}{cccc}1& 0& 3& 5\\ 0& 1& -2& 2\end{array}\right]$$

• Then we can write the solution vector as

$$\begin{array}{r}z=t,x=5-3t,y=2+2t\\ \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}5\\ 2\\ 0\end{array}\right]+t\left[\begin{array}{c}-2\\ 2\\ 1\end{array}\right]\end{array}(line\in {\mathbb{R}}^3)$$