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?
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.