Finding Value of k for No Solution

Video Explanation

Question

Find the value of \(k\) for which the system of equations \(x + 2y – 3 = 0\) and \(5x + ky + 7 = 0\) has no solution.

Solution

Step 1: Identify Coefficients

For equation (1): \(a_1 = 1,\; b_1 = 2,\; c_1 = -3\)

For equation (2): \(a_2 = 5,\; b_2 = k,\; c_2 = 7\)

Step 2: Apply Condition for No Solution

For no solution:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \]

\[ \frac{1}{5} = \frac{2}{k} \]

Step 3: Solve

\[ k = 10 \]

Step 4: Check Condition

\[ \frac{c_1}{c_2} = \frac{-3}{7} \ne \frac{1}{5} \]

⇒ Condition satisfied ✔

Final Answer

\[ \text{The system has no solution when } k = 10. \]