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

Step 1: Condition for no solution:
Linear equations have no solution if their graphs are parallel lines. Parallel lines have equal slopes but different y-intercepts.
The general form of a linear equation is \( Ax + By + C = 0 \), with slope \( m = -\frac{A}{B} \).
For no solution, the slopes must be equal, and the lines must not be identical.

Step 2: Convert equations to slope-intercept form:
Equation 1: \[ 5x + 2y - 7 = 0 \] Rearranging to \(y = mx + c\): \[ 2y = -5x + 7 \quad \Rightarrow \quad y = -\frac{5}{2}x + \frac{7}{2} \] Slope \( m_1 = -\frac{5}{2} \).

Equation 2: \[ 2x + ky + 1 = 0 \] Rearranging to \( y = mx + c \): \[ ky = -2x - 1 \quad \Rightarrow \quad y = -\frac{2}{k}x - \frac{1}{k} \] Slope \( m_2 = -\frac{2}{k} \).

Step 3: Equate slopes for parallel lines:
For parallel lines, \( m_1 = m_2 \): \[ -\frac{5}{2} = -\frac{2}{k} \] Solving for \(k\): \[ \frac{5}{2} = \frac{2}{k} \quad \Rightarrow \quad 5k = 4 \quad \Rightarrow \quad k = \frac{4}{5} \]

Step 4: Conclusion:
The value of \( k \) for which the system has no solution is \( k = \frac{4}{5} \).