The system of linear equations can be written in matrix form as AX = B, where:
A = [1 1 -1
2 3 1
3 -1 -7]
X = [x
y
z]
B = [3
10
1]
We can use matrix elimination or Gaussian elimination to solve for X:
Step 1: Subtract 2 times the first row from the second row:
A = [1 1 -1
0 -1 3
3 -1 -7]
B = [3
4
1]
Step 2: Subtract 3 times the first row from the third row:
A = [1 1 -1
0 -1 3
0 -4 -20]
B = [3
4
-8]
Step 3: Divide the second row by -1:
A = [1 1 -1
0 1 -3
0 -4 -20]
B = [3
-4
-8]
Step 4: Subtract the second row from the third row:
A = [1 1 -1
0 1 -3
0 0 -16]
B = [3
-4
-12]
Step 5: Divide the third row by -16:
A = [1 1 -1
0 1 -3
0 0 1]
B = [3
-4
12/16]
Step 6: Subtract -3 times the second row from the first row:
A = [1 0 4
0 1 -3
0 0 1]
B = [15/4
-4
12/16]
Step 7: Solve for x by multiplying the first row by 4:
A = [1 0 4
0 1 -3
0 0 1]
B = [3
-4
12/16]
X = [3
-4
3/4]
So the solution to the system of linear equations is X = [3, -4, 3/4].