Solution set of x + y + z = 2, x + y + z = 5

Step by step

The question

Question: find all the solutions of x + y + z = 2 and x + y + z = 5.

We want every x, y and z that make both true.

There are 3 letters, but only 2 equations.

1 Write the augmented matrix

Each row is one equation: the numbers in front of x, y and z, 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.

Row 2 is all 0s, then 3.

It says 0 = 3, which is false.

So there is no solution: the solution set is empty.

Let's look back.

First: write the augmented matrix.

Last: row reduce.