Answer:
(x, y, z) = (18/7, 5/7, -16/5)
Explanation:
To solve the matrix, we can use the method of matrix row operations to bring it to its row-echelon form or reduced row-echelon form. Here, we'll use the row-echelon form.
Starting matrix:
3 7 1
2 -4 -4
-6 -15 -12
-4 -3 15
16
Divide the first row by 3:
1 7/3 1/3
2 -4 -4
-6 -15 -12
-4 -3 15
16
Multiply the first row by -2 and add it to the second row:
1 7/3 1/3
0 -14/3 -10/3
-6 -15 -12
-4 -3 15
16
Multiply the first row by 6 and add it to the third row:
1 7/3 1/3
0 -14/3 -10/3
0 -7 0
-4 -3 15
16
Multiply the first row by 4 and add it to the fourth row:
1 7/3 1/3
0 -14/3 -10/3
0 -7 0
0 7 0
16
Divide the second row by -14/3 (or multiply by -3/14):
1 7/3 1/3
0 1 5/7
0 -7 0
0 7 0
16
Add 7/3 times the second row to the first row:
1 0 18/7
0 1 5/7
0 -7 0
0 7 0
16
Add 7 times the second row to the third row:
1 0 18/7
0 1 5/7
0 0 0
0 7 0
16
Subtract 7 times the second row from the fourth row:
1 0 18/7
0 1 5/7
0 0 0
0 0 -5
16
Divide the fourth row by -5:
1 0 18/7
0 1 5/7
0 0 0
0 0 1
-16/5
Now, the matrix is in row-echelon form, and you can read the solution:
x = 18/7
y = 5/7
z = -16/5
So, the solution to the matrix is (x, y, z) = (18/7, 5/7, -16/5).