For a certain value of k, the system

x + y + 3z = 10,

-4x + 2y + 5z = 7,

kx + z = 3

has no solutions. What is this value of k?

I'd start by calculating the determinant of the coefficient matrix. That's a cubic in k. Then find the values of k that make this determinant 0.

When the determinant is not 0, the system has a unique solution.

When the determinant is 0, the system is linearly dependent and has infinite solutions, or is inconsistent and has no solutions. Which option depends on the constants on the right.

So take each value of k that makes the system 0, and see whether the equations are identical or inconsistent.