Solve x + y = 5, x - y = 1 by gaussian elimination Step by Step

Step by step

The question

Question: solve x + y = 5 and x − y = 1 by Gaussian elimination.

x + y = 5①

We want the x and y that make both true.

x − y = 1②

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, 1, 5; 1, −1, 1]

We call the rows R 1 and R 2.

Our goal: 1s down the diagonal, and 0s under them.

This shape is called row-echelon form.

Then we go back up, one letter at a time.

goal: 1s down the diagonal, 0s under them

2 Get 0s under the 1 in column 1

→ [1, 1, 5; 0, −2, −4]

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

3 Get a 1 in row 2

→ [1, 1, 5; 0, 1, 2]

4 Back substitution

Start at the bottom.

Each row is an equation again.

Row 2 is 1y = 2, with 0 for the other letters.

So y = 2.

y = 2

Row 1 is the equation x + y = 5.

x + y = 5

We know y = 2.

We put it in.

x + 2 = 5

Work it out: x + 2 = 5.

x + 2 = 5

To remove the + 2, we take away 2 on both sides.

x + 2 − 2 = 5 − 2

So x = 3.

x = 3

We write it as (3, 2).

(x, y) = (3, 2)

5 Check

Let's check in every equation.

We put x = 3, y = 2 into ①.

① 3 + 2 = 5

We get 5, the same as the right side.

Now ②.

② 3 − 2 = 1

We get 1, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: write the augmented matrix.

Then: get 0s under the 1 in column 1.

Then: get a 1 in row 2.

Then: back substitution.

Last: check.

Answer