If 3x − ky + 8 = 0 and 9x − 18y + 20 = 0 have no solution, find k.

Question

If 3x − ky + 8 = 0 and 9x − 18y + 20 = 0 have no solution, find k.

1. A. 2
2. B. 6 (Correct answer)
3. C. 9
4. D. 10

Correct Answer

Option B — 6

Detailed Solution & Explanation

The correct answer is 6.

Key Points

• Two linear equations have no solution when their lines are parallel — the coefficients are proportional but the constants are not:
  • a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Comparing 3x − ky + 8 = 0 with 9x − 18y + 20 = 0:
  • 3/9 = (−k)/(−18), so 1/3 = k/18
  • k = 6
• Check the inequality: c₁/c₂ = 8/20 = 2/5, which differs from 1/3, so the lines really are parallel and not identical.

Additional Information

• The three cases for a pair of linear equations:
  • a₁/a₂ ≠ b₁/b₂ — intersecting lines, a unique solution (consistent)
  • a₁/a₂ = b₁/b₂ = c₁/c₂ — coincident lines, infinitely many solutions
  • a₁/a₂ = b₁/b₂ ≠ c₁/c₂ — parallel lines, no solution (inconsistent)
• Verifying the third ratio matters: had 8/20 also equalled 1/3, the answer would be "infinitely many solutions" rather than none.
• Equivalently, no solution means equal slopes with different intercepts — here both slopes come to 1/2.