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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?

 Mar 6, 2023
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We can solve this system of equations using Gaussian elimination. First, we can add 4 times the first equation to the second equation to eliminate the x term:

x + y + 3z = 10
(4)(x + y + 3z) + (-4x + 8y + 5z) = (4)(10) + 7
kx + z = 3

Simplifying, we get:

x + y + 3z = 10
20y + 17z = 47
kx + z = 3

Next, we can subtract 20 times the third equation from the second equation to eliminate the y term:

x + y + 3z = 10
-20kx - 20z + 17z = 47 - 60k
kx + z = 3

Simplifying, we get:

x + y + 3z = 10
(k - 3)z = 60k - 47
kx + z = 3

If this system has no solutions, then the equations must be inconsistent, which means that there is no value of k that satisfies all three equations. 

In particular, the second equation cannot be satisfied for any value of k, since the left-hand side is always a multiple of 17, while the right-hand side is not a multiple of 17.

Therefore, we can set the coefficients of z equal to each other and solve for k:

k - 3 = 60k - 47

Simplifying, we get:

59k = 50

k = 50/59

So the value of k that makes the system inconsistent is k = 50/59.

 Mar 6, 2023

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