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

Problem:
Given two linear equations in two variables:
1) \( 5x + 2y - 7 = 0 \)
2) \( 2x + ky + 1 = 0 \)
Determine the value of \(k\) for which these equations are inconsistent (have no solution).

Step 1: Condition for no solution
For linear equations in the form \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\), the system has no solution if:
\[\frac{a_1}{a_2} = \frac{b_1}{b_2} e \frac{c_1}{c_2}\]

Step 2: Identify coefficients
From equation 1: \( a_1 = 5, b_1 = 2, c_1 = -7 \)
From equation 2: \( a_2 = 2, b_2 = k, c_2 = 1 \)

Step 3: Apply the condition \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \)
\[\frac{5}{2} = \frac{2}{k}\]
Solving for \(k\):
\[5k = 4\]
\[k = \frac{4}{5}\]

Step 4: Verify the condition \( \frac{b_1}{b_2} e \frac{c_1}{c_2} \)
Substitute \(k = \frac{4}{5}\) and check if:
\[\frac{2}{4/5} e \frac{-7}{1}\]
\[\frac{10}{4} e -7\]
\[\frac{5}{2} e -7 \quad \text{(This is true)}\]

Final Answer:
The value of \(k\) that results in no solution is \(\frac{4}{5}\).