Question

Find the complete solution of the linear system, or show that it is inconsistent. {x-4 z=1 {2 x-y-6 z=4 {2 x+3 y-2 z=8

Answer 1

To find the complete solution of the linear system or determine if it is inconsistent, we can use the method of Gaussian elimination or row reduction. Let's write the system of equations in augmented matrix form:

[1 -4 0 | 1]
[2 -1 -6 | 4]
[2 3 -2 | 8]

Performing row operations to simplify the matrix:

R2 = R2 - 2R1
R3 = R3 - 2R1

[1 -4 0 | 1]
[0 7 -6 | 2]
[0 11 -2 | 6]

R3 = R3 - (11/7)R2

[1 -4 0 | 1]
[0 7 -6 | 2]
[0 0 2 | 2]

Now, let's convert the matrix back into equation form:

x - 4z = 1
7y - 6z = 2
2z = 2

From the third equation, we can see that z = 1. Substituting this value into the second equation:

7y - 6(1) = 2
7y - 6 = 2
7y = 8
y = 8/7

Finally, substituting the values of y and z into the first equation:

x - 4(1) = 1
x - 4 = 1
x = 5

Therefore, the complete solution to the linear system is:
x = 5
y = 8/7
z = 1

The system is consistent, and the solution is unique.

Answer 2

The linear system of equations is inconsistent and has no solution.

Step-by-step explanation:

To solve this linear system of equations, we can use the method of elimination or substitution. In this case, let's use the method of elimination.

First, we'll eliminate the variable 'x' by multiplying the first equation by 2 and subtracting it from the second equation. This gives us:

-2x - 6z = -6

Next, we'll eliminate 'x' again by multiplying the first equation by 2 and subtracting it from the third equation. This gives us:

-2x - 6z = 6

Now we have two equations in terms of 'z'. Subtracting these equations will give us:

0 = 12

Since 0 is not equal to 12, this means the system of equations is inconsistent and has no solution.