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For a certain value of k, the system

x + y + 3z = 10

-4x + 8y + 5z = 7

kx + z = 3

has no solutions.  What is this value of k?

 Feb 10, 2023
 #1
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This system of equations can be written as an augmented matrix and transformed into an echelon form or reduced row-echelon form using row operations. If the reduced row-echelon form has a row of zeros except for the last entry (which is non-zero), then the system has no solutions.

Here's the augmented matrix of the system:

[1 1 3 10]

[-4 8 5 7]

[k 0 1 3]

Using row operations, we can transform the matrix into row-echelon form:

[1 1 3 10]

[0 9 2 11]

[0 0 1-k 3-3k]

From the last row, we see that the system has no solutions if and only if 1-k is non-zero and 3-3k is non-zero. That is, if 3k ≠ 3 and k ≠ 1.

So the value of k that makes the system have no solutions is k = 1.

 Feb 10, 2023

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