For what value of k does the system of equations have no solution? 4x − 8y = 3 2x − ky = 5

Question

For what value of k does the system of equations have no solution?

4x − 8y = 3

2x − ky = 5

Answer 1

Another way to answer this kind of question is to set the principal determinant of the system equal to zero, and then solve for k.

|4 -8|

|2 -k| = 0

-4k - 2(-8) = 0

-4k + 16 = 0

k = 4

Answer 2

There is no solution when the two lines a parallel meaning they have the same slope. To find the slope of the first line, put it in slope-intercept form (y = mx + b)

4x − 8y = 3

4x - 3 = 8y

y = 4/8x - 3/8 or y = 1/2x - 38

So the slope is 1/2

Now do the same for the 2nd equation:

2x − ky = 5

2x - 5 = ky

y = (2/k)x - 5/k

So the slope is 2/k

But we need to have 2/k = 1/2 for the slopes to be equal so k must be 4