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 \(\frac{4}{5}\).