Solution set of x + y = 3, x - y = 1 Step by Step

Step by step

The question

Question: find all the solutions of x + y = 3 and x − y = 1.

x + y = 3①

We want every x and y that make both true.

x − y = 1②

1 Write the augmented matrix

Each row is one equation: the numbers in front of x and y, then the number on the right, after the bar.

= [1, 1, 3; 1, −1, 1]

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.

swap two rows

× a row by a number (not 0)

add a multiple of one row to another

→ [1, 1, 3; 0, −2, −2]

→ [1, 1, 3; 0, 1, 1]

→ [1, 0, 2; 0, 1, 1]

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.

Every letter has a leading 1.

There is no free letter, so there is just one solution.

leading 1: x, y

4 Read the answer

Row 1 says x = 2.

x = 2

Row 2 says y = 1.

y = 1

One solution: (2, 1).

[x; y] = [2; 1]

5 Check

Let's check: we put x = 2 and y = 1 into every equation.

① 2 + 1 = 3

② 2 − 1 = 1

Every equation works.

So the answer is right!

Let's look back.

First: write the augmented matrix.

Then: row reduce.

Then: find the free letters.

Then: read the answer.

Last: check.

Answer