For what value of k does the system of equations kx - 15y + 7 = 0 and 7x - 21y - 19 = 0 have no solution?

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Given the system of equations:
1) kx - 15y + 7 = 0
2) 7x - 21y - 19 = 0

To check when the system has no solution, the lines must be parallel but not coincident.

Rewrite both equations in the form Ax + By + C = 0:
Equation 1: kx - 15y + 7 = 0
Equation 2: 7x - 21y - 19 = 0

For two lines A1x + B1y + C1 = 0 and A2x + B2y + C2 = 0,
- They are parallel if A1/A2 = B1/B2 ≠ C1/C2

Calculate ratios:
A1/A2 = k/7
B1/B2 = (-15)/(-21) = 15/21 = 5/7

Set A1/A2 = B1/B2:

k/7 = 5/7
=> k = 5

Now check if C1/C2 ≠ A1/A2:
C1/C2 = 7/(-19) = -7/19

Since k/7 = 5/7 = 5/7 and C1/C2 = -7/19, they are not equal.

Hence, for k = 5, the lines are parallel but not coincident, so the system has no solution.

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