Step by step
The question
Question: find all the solutions of x + 2y − z + w = 3 and y + z − w = 1.
We want every x, y, z and w that make both true.
There are 4 letters, but only 2 equations.
1 Write the augmented matrix
Each row is one equation: the numbers in front of x, y, z and w, then the number on the right, after the bar.
We call the rows R 1 and R 2.
2 Row reduce
We use the three row moves: swap two rows, times a row by a number that is not 0, add a multiple of one row to another.
They never change the answers.
Our goal: a leading 1 in each row, with 0s above and below it.
3 Find the free letters
The first number that is not 0 in each row is a 1.
We call it a leading 1, and ring it.
The letters x and y have a leading 1 in their columns.
The letters z and w have none: they are free.
A free letter can be any number.
4 Write the leading letters with the free ones
Row 1 as an equation: x − 3z + 3w = 1.
We want x alone.
To remove − 3z and + 3w, we add 3z and take away 3w on both sides.
They cancel on the left.
So x = 1 + 3z − 3w.
Careful!
The − 3z moves to the other side, so its sign changes.
So x is not 1 − 3z.
Row 2 as an equation: y + z − w = 1.
We want y alone.
To remove + z and − w, we take away z and add w on both sides.
They cancel on the left.
So y = 1 − z + w.
5 Write it as vectors
We stack the letters in a column.
The free letters stay as they are.
We split it up.
The numbers alone make one vector: (1, 1, 0, 0).
What goes with z makes another: (3, −1, 1, 0), and the same for w.
So every solution is one point, plus any amount of each direction.
Two free letters: the solutions make a flat plane.
6 Check
Let's check.
First the point: all free letters 0.
We put x = 1, y = 1, z = 0 and w = 0 into every equation.
The point works in every equation.
Now the direction (3, −1, 1, 0).
Its left sides must give 0, so adding it changes nothing.
Now the direction (−3, 1, 0, 1).
Its left sides must give 0, so adding it changes nothing.
Every left side gives 0.
So adding any amount of it keeps every equation true.
The answer is right!
Let's look back.
First: write the augmented matrix.
Then: row reduce.
Then: find the free letters.
Then: write the leading letters with the free ones.
Then: write it as vectors.
Last: check.