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:
We are given two linear equations in two variables:
1) \( 5x + 2y - 7 = 0 \)
2) \( 2x + ky + 1 = 0 \)
We are asked to find the value of \(k\) for which these equations do not have a solution.

Step 1: Understand when a pair of linear equations has no solution
For two linear equations in the form:
\[ a_1x + b_1y + c_1 = 0 \quad \text{and} \quad a_2x + b_2y + c_2 = 0 \]
The condition for no solution (i.e., inconsistent system) is:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \]

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 for no solution
Use the first part of the inconsistency condition:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \Rightarrow \frac{5}{2} = \frac{2}{k} \]
Now solve for \(k\):
\[ \frac{5}{2} = \frac{2}{k} \Rightarrow 5k = 4 \Rightarrow k = \frac{4}{5} \]

Step 4: Check the second part of the condition
To confirm that the system has no solution, we must ensure:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \Rightarrow \frac{5}{2} \ne \frac{-7}{1} \quad \text{which is clearly true} \]

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