Find a system of linear equations corresponding to the following augmented matrices:
(a)
2 0 |0
3 −4 |0
0 1 |1
(b)
3 0 −2 |5
7 1 4 |−3
0 −2 1 |7
(c)
7 2 1 −3 |5
1 2 4 0 |1
(d)
1 0 0 0 |7
0 1 0 0 |−2
0 0 1 0 |3
0 0 0 1 |4
To find the system of linear equations corresponding to each augmented matrix, we need to write the augmented matrix in the form of a system of linear equations.
(a)
The augmented matrix
2 0 |0
3 −4 |0
0 1 |1
corresponds to the system of linear equations:
2x + 0y = 0
3x - 4y = 0
0x + 1y = 1
Therefore, the system of linear equations is:
2x = 0
3x - 4y = 0
y = 1
(b)
The augmented matrix
3 0 −2 |5
7 1 4 |−3
0 −2 1 |7
corresponds to the system of linear equations:
3x + 0y - 2z = 5
7x + y + 4z = -3
0x - 2y + z = 7
Therefore, the system of linear equations is:
3x - 2z = 5
7x + y + 4z = -3
-2y + z = 7
(c)
The augmented matrix
7 2 1 −3 |5
1 2 4 0 |1
corresponds to the system of linear equations:
7x + 2y + z - 3w = 5
1x + 2y + 4z + 0w = 1
Therefore, the system of linear equations is:
7x + 2y + z - 3w = 5
1x + 2y + 4z = 1
(d)
The augmented matrix
1 0 0 0 |7
0 1 0 0 |−2
0 0 1 0 |3
0 0 0 1 |4
corresponds to the system of linear equations:
1x + 0y + 0z + 0w = 7
0x + 1y + 0z + 0w = -2
0x + 0y + 1z + 0w = 3
0x + 0y + 0z + 1w = 4
Therefore, the system of linear equations is:
x = 7
y = -2
z = 3
w = 4
This is how you find the system of linear equations corresponding to an augmented matrix.