The question
Question: solve x + 2y = 4 and x + y = 3 with a matrix.
x + 2y = 4①
We want the x and y that make both true.
x + y = 3②
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, 4; 1, 1, 3]
We call the rows R 1 and R 2.
2 Get 0s in column 1
→ [1, 2, 4; 0, −1, −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
3 Get a 1 in row 2
→ [1, 2, 4; 0, 1, 1]
4 Get 0s in column 2
→ [1, 0, 2; 0, 1, 1]
5 Read the answer
Each row is an equation again.
Row 1 is 1x + 0y = 2.
So x = 2.
x = 2
Row 2: y = 1.
y = 1
We write it as (2, 1).
(x, y) = (2, 1)
6 Check
Let's check in every equation.
We put x = 2, y = 1 into ①.
① 2 + 2(1) = 2 + 2 = 4
We get 4, the same as the right side.
Now ②.
② 2 + 1 = 3
We get 3, 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 in column 1.
Then: get a 1 in row 2.
Then: get 0s in column 2.
Then: read the answer.
Last: check.
▶ Watch it solved
The question
Question: solve −x + y = 0 and −3x + 4y = 2 with a matrix.
−x + y = 0①
We want the x and y that make both true.
−3x + 4y = 2②
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, 0; −3, 4, 2]
We call the rows R 1 and R 2.
2 Get a 1 in row 1
→ [1, −1, 0; −3, 4, 2]
3 Get 0s in column 1
→ [1, −1, 0; 0, 1, 2]
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
4 Get 0s in column 2
→ [1, 0, 2; 0, 1, 2]
5 Read the answer
Each row is an equation again.
Row 1 is 1x + 0y = 2.
So x = 2.
x = 2
Row 2: y = 2.
y = 2
We write it as (2, 2).
(x, y) = (2, 2)
6 Check
Let's check in every equation.
We put x = 2, y = 2 into ①.
① −2 + 2 = 0
We get 0, the same as the right side.
Now ②.
② −3(2) + 4(2) = −6 + 8 = 2
We get 2, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: write the augmented matrix.
Then: get a 1 in row 1.
Then: get 0s in column 1.
Then: get 0s in column 2.
Then: read the answer.
Last: check.
▶ Watch it solved
The question
Question: solve x + 2y + z = 5, x + 3y + 3z = 2 and 2x + y − 3z = 17 with a matrix.
x + 2y + z = 5①
x + 3y + 3z = 2②
We want the x, y and z that make all three true.
2x + y − 3z = 17③
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, y and z, then the number on the right.
The bar stands for the = signs.
This is called an augmented matrix.
= [1, 2, 1, 5; 1, 3, 3, 2; 2, 1, −3, 17]
We call the rows R 1, R 2 and R 3.
2 Get 0s in column 1
→ [1, 2, 1, 5; 0, 1, 2, −3; 0, −3, −5, 7]
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 0s in column 2
→ [1, 0, −3, 11; 0, 1, 2, −3; 0, 0, 1, −2]
4 Get 0s in column 3
→ [1, 0, 0, 5; 0, 1, 0, 1; 0, 0, 1, −2]
5 Read the answer
Each row is an equation again.
Row 1 is 1x + 0y + 0z = 5.
So x = 5.
x = 5
Row 2: y = 1.
y = 1
Row 3: z = −2.
z = −2
We write it as (5, 1, −2).
(x, y, z) = (5, 1, −2)
6 Check
Let's check in every equation.
We put x = 5, y = 1, z = −2 into ①.
① 5 + 2(1) + (−2) = 5 + 2 − 2 = 5
We get 5, the same as the right side.
Now ②.
② 5 + 3(1) + 3(−2) = 5 + 3 − 6 = 2
We get 2, the same as the right side.
Now ③.
③ 2(5) + 1 − 3(−2) = 10 + 1 + 6 = 17
We get 17, the same as the right side.
All of them work.
So the answer is right!
Let's look back.
First: write the augmented matrix.
Then: get 0s in column 1.
Then: get 0s in column 2.
Then: get 0s in column 3.
Then: read the answer.
Last: check.
Answer (x, y, z) = (5, 1, −2)
▶ Watch it solved
