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

Answer: (B)

$\frac{4}{5}$

Answer 2

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 $\frac{4}{5}$.