Find k if the following system of equations has no solution 3x + 2y = 6 and kx + (k -1)y = 9?

Answer 1

If 3x + 2y = 6 then y = (-3/2)x + 3. This is the equation of a line with slope -3/2.
If kx + (k-1)y = 9 then y = (-k/(k-1))x + 9/(k-1). This is the equation of a line with slope -k/(k-1).
If the system has no solution then the lines must be parallel, i.e. they must have the same slope. Otherwise they will intersect, and their intersection is a solution.
So -3/2 = -k/(k-1), or 2k = 3(k-1) = 3k - 3. Then 3k - 2k = 3, or k = 3.

The system has no solution iff k = 3.

Answer 2

I can see by inspection that k=3 will do, but let's justify the answer.

If the system of equations has no solution, it represents parallel lines. Parallel lines have the same slope, but different y-intercepts. Write both equations in y=mx+b form to find their slopes:

3x+2y=6
2y=6-3x
y=(-3/2)x+3

kx+(k-1)y=9
(k-1)y=-kx+9
y=(-k/(k-1))x+9/(k-1)

Make the two slopes equal to each other:

-3/2 = -k/(k-1)
3(k-1) = 2k
3k-3 = 2k
k = 3

Check in the original equation. Inserting k=3 gives a second equation 3x+2y=9, which is clearly parallel to 3x+2y=6.

Answer 3

'No solution' means the equations represent parallel lines >> coefficients of x and y are proportional >> slopes are same

k/3 = (k-1)/2
Solve for k
k = 3