For what value of k, the following system of equations has infinitely many solutions?
\(x+2y= 5, 3x+ky = 15\)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 6
A system with a unique solution has \(|A| \ne 0\), so infinitely many solutions need the coefficient determinant to be 0:
\( |A| = \begin{vmatrix}1&2\\3&k\end{vmatrix} = k - 6 \)
\(k - 6 = 0,\quad k = 6\)
With \(k = 6\), the second equation is
\(3x + 6y = 15\), that is, \(3(x + 2y) = 3 \times 5\)
which is the first equation multiplied by 3. The two equations are the same line, so every point on \(x + 2y = 5\) is a solution.