The system of linear equations \(kx + 5y = 5, 2x + 3y = 5\) will be consistent if
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(k \ne \displaystyle\frac{10}{3}\)
A system is consistent if it has at least one solution. It has a unique solution when the coefficient determinant is non-zero:
\( \displaystyle \begin{vmatrix}k&5\\2&3\end{vmatrix} = 3k - 10 \)
\( \displaystyle 3k - 10 \ne 0 \iff k \ne \frac{10}{3} \)
When \(k = \frac{10}{3}\), multiplying the first equation by \(\frac{3}{5}\) gives:
\(2x + 3y = 3\)
This contradicts \(2x + 3y = 5\), so there is no solution. Hence the system is consistent exactly when \(k \ne \frac{10}{3}\).