Science:Math Exam Resources/Courses/MATH152/April 2022/Question A11/Solution 2

Alternatively, this problem can be solved simply by solving the system equations we solved to find the point ${\textstyle p}$ in the previous solution. To recap, that system is

$$\left[{\begin{array}{ccc|c}1&-2&1&1\\1&-1&2&2\end{array}}\right].$$

Because this is a system with more variables than equations, it will either have no solution, or infinitely many solutions. Looking at the normal vectors for the planes, we can see that they are not parallel, meaning that the planes will intersect along a line!

Recall that the system simplified to

$${\begin{aligned}&x_{1}=3(1-x_{3})\\&x_{2}=1-x_{3}\end{aligned}}.$$

Now, since ${\textstyle x_{3}}$ is a free variable, let ${\textstyle x_{3}=t}$, giving the system

$$\left\lbrace {\begin{aligned}&x_{1}=3(1-t)\\&x_{2}=1-t\\&x_{3}=t\end{aligned}}\right.\rightarrow \left[{\begin{array}{c}x_{1}\\x_{2}\\x_{3}\end{array}}\right]=\left[{\begin{array}{c}3\\1\\0\end{array}}\right]+t\left[{\begin{array}{c}-3\\-1\\1\end{array}}\right].$$