Step by step
The question
Question: solve x + 2y = 3 and 2x + 4y = 5 with a matrix.
x + 2y = 3①
We want the x and y that make both true.
2x + 4y = 5②
We will use three moves on the rows.
Swap two rows.
Times a row by a number that is not 0.
Add a multiple of one row to another.
Each move keeps the equations true, like in elimination.
swap two rows
× a row by a number (not 0)
add a multiple of one row to another
1 Write the augmented matrix
Each row is one equation: the numbers in front of x and y, then the number on the right.
The bar stands for the = signs.
This is called an augmented matrix.
= [1, 2, 3; 2, 4, 5]
We call the rows R 1 and R 2.
2 Get 0s in column 1
→ [1, 2, 3; 0, 0, −1]
Careful!
A row move works on the whole row, the number after the bar too.
That number is the right side of the equation.
Careful: the whole row, after the bar too
One row is all 0s, then −1.
It says 0 = −1, which is never true.
So there is no solution.
0 = −1: no solution
Let's look back.
First: write the augmented matrix.
Last: get 0s in column 1.
Answer
0 = −1: no solution