Solve x + 3y = 7, 2x + y = 4 by gaussian elimination Step by Step

Step by step

The question

Question: solve x + 3y = 7 and 2x + y = 4 by Gaussian elimination.

x + 3y = 7①

We want the x and y that make both true.

2x + y = 4②

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, 3, 7; 2, 1, 4]

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, 3, 7; 0, −5, −10]

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, 3, 7; 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 + 3y = 7.

x + 3y = 7

We know y = 2.

We put it in.

x + 3 × 2 = 7

Work it out: x + 6 = 7.

x + 6 = 7

To remove the + 6, we take away 6 on both sides.

x + 6 − 6 = 7 − 6

So x = 1.

x = 1

We write it as (1, 2).

(x, y) = (1, 2)

5 Check

Let's check in every equation.

We put x = 1, y = 2 into ①.

① 1 + 3(2) = 1 + 6 = 7

We get 7, the same as the right side.

Now ②.

② 2(1) + 2 = 2 + 2 = 4

We get 4, 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