Question

Use row operations to solve the system.

x+y−z=4
4x−z+z=6
x−3y+2z=-16
Select the correct choice below and fill in any answer boxes within your choice.
A. There is one solution. The solution is (Type an exact answer in simplified form.)
B. There are infinitely many solutions. The solutions are (_,_,z). where z is any real number.
C. There is no solution.

Answer 1

To solve the system of equations using row operations, create an augmented matrix and perform row operations to simplify the matrix. The solution to the system is x = 1, y = 2, and z = -6/7.

Step-by-step explanation:

To solve the system of equations using row operations, we can create an augmented matrix to represent the system:

[1 1 -1 | 4]

[4 0 1 | 6]

[1 -3 2 | -16]

Next, we can perform row operations to simplify the matrix:

R2 = R2 - 4R1

[1 1 -1 | 4]

[0 -4 5 | -10]

[1 -3 2 | -16]

R3 = R3 - R1

[1 1 -1 | 4]

[0 -4 5 | -10]

[0 -4 3 | -12]

R3 = R3 + (4/5)R2

[1 1 -1 | 4]

[0 -4 5 | -10]

[0 0 7 | -6]

Finally, we can divide the third row by 7 to get:

[1 1 -1 | 4]

[0 -4 5 | -10]

[0 0 1 | -6/7]

From the matrix, we can see that there is one solution. The solution is x = 1, y = 2, and z = -6/7. Therefore, the correct choice is A. There is one solution. The solution is (1, 2, -6/7).