Question

The value of \(k\) for which the pair of linear equations \(5x + 2y - 7 = 0\) and \(2x + ky + 1 = 0\) don't have a solution, is:

• A. 5
• B. \(\frac{4}{5}\)
• C. \(\frac{5}{4}\)
• D. \(\frac{5}{2}\)

Answer 1

The Correct Option is B

For a pair of linear equations to have no solution, the condition is that the determinant of the coefficient matrix should be zero. The general form of two linear equations is:

\[ a_1x + b_1y + c_1 = 0 \quad \text{and} \quad a_2x + b_2y + c_2 = 0 \]

For the given equations:

1) \(5x + 2y - 7 = 0\) \quad 2) \(2x + ky + 1 = 0\)

The coefficient matrix is:

\[ \begin{pmatrix} 5 & 2 \\ 2 & k \end{pmatrix} \]

The determinant of the coefficient matrix is:

\[ \text{Determinant} = (5)(k) - (2)(2) = 5k - 4 \]

For no solution, the determinant must be zero:

\[ 5k - 4 = 0 \]

Solving for \(k\):

\[ 5k = 4 \implies k = \frac{4}{5} \]

Thus, the value of \(k\) for which the pair of equations has no solution is \(\frac{4}{5}\).