Linear algebra, common points between 3 planes

Question

Find all the common points between the planes x+2y+z=1, 2x+4y+z=3 andx+2y+3z=−1

can you guys explain all steps im so stupid in linear thank you

Answer 1

This is the non homogeneous linear system $$ \left( \begin{array}{ccc} 1 & 2 & 1 \\ 2 & 4 & 1 \\ 1 & 2 & 3 \end{array} \right) \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} 1 \\ 3 \\ -1 \end{array} \right) $$ which can be written as $$ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 1 \\ 2 & 4 & 1 & 3 \\ 1 & 2 & 3 & -1 \end{array} \right] $$ and solved by Gauss elimination. Subtracting two times the first from the second row and subtracting the first from the third row gives $$ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 1 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 2 & -2 \end{array} \right] $$ which is equivalent to $$ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array} \right] \to \left[ \begin{array}{ccc|c} 1 & 2 & 0 & 2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array} \right] $$ And this means $$ x + 2 y = 2 \\ z = -1 $$ which is the line $$ L = \{ (x, -(1/2) x + 1, -1) \mid x \in \mathbb{R} \} $$

Answer 2

You have to write the augmented matrix of the linear system and pˆroced with Gauß elimination to obtain the rediced row d=reduction form of the matrix of the left-hand side: \begin{align} &\begin{bmatrix}\begin{array}{ccc|r} 1&2&1&1\\ 2&4&1&3\\ 1&2&3&-1 \end{array} \end{bmatrix}\rightsquigarrow \begin{bmatrix}\begin{array}{ccc|rc} 1&2&1&1\\ 0&0&-1&1\\ 0&0&2&-2 \end{array} \end{bmatrix}\rightsquigarrow \begin{bmatrix}\begin{array}{ccc|rc} 1&2&1&1\\ 0&0&-1&1\\ 0&0&0&0 \end{array} \end{bmatrix}\\[1ex] \rightsquigarrow&\begin{bmatrix}\begin{array}{ccc|rc} 1&2&0&2\\ 0&0&1&-1\\ 0&0&0&0 \end{array} \end{bmatrix} \end{align}

So the solutions are an affine subspace of dimension $1$ ( a line): $$x+2y=2,\quad z=-1$$ or, in vector form, setting $y=t$: $$\begin{bmatrix}x\\y\\z \end{bmatrix}= \begin{bmatrix}2\\0\\-1 \end{bmatrix}+t \begin{bmatrix}-2\\1\\0 \end{bmatrix}.$$