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: Understanding the condition for no solution:
For a pair of linear equations to have no solution, the lines represented by the equations must be parallel. Two lines are parallel if their slopes are equal but their intercepts are different.
The general form of a linear equation is \( Ax + By + C = 0 \), and the slope of the line is given by \( m = -\frac{A}{B} \).
Thus, for the given pair of equations to have no solution, the slopes of the two lines must be equal, but the lines should not coincide.

Step 2: Writing the given equations in slope-intercept form:
The first equation is: \[ 5x + 2y - 7 = 0 \] Rearranging this to the form \(y = mx + c\), we get: \[ 2y = -5x + 7 \quad \Rightarrow \quad y = -\frac{5}{2}x + \frac{7}{2} \] The slope of this line is \( m_1 = -\frac{5}{2} \).

The second equation is: \[ 2x + ky + 1 = 0 \] Rearranging this to the form \( y = mx + c \), we get: \[ ky = -2x - 1 \quad \Rightarrow \quad y = -\frac{2}{k}x - \frac{1}{k} \] The slope of this line is \( m_2 = -\frac{2}{k} \).

Step 3: Setting the slopes equal for parallel lines:
For the lines to be parallel, their slopes must be equal. Therefore, we set \( m_1 = m_2 \):
\[ -\frac{5}{2} = -\frac{2}{k} \] Now, solve 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 pair of linear equations do not have a solution is \( k = \frac{4}{5} \).