Finding Value of k for No Solution

Video Explanation

Question

Find the value of \(k\) for which the system of equations \(kx – 5y = 2\) and \(6x + 2y = 7\) has no solution.

Solution

Step 1: Write in Standard Form

\[ kx – 5y – 2 = 0 \]

\[ 6x + 2y – 7 = 0 \]

Step 2: Identify Coefficients

\(a_1 = k,\; b_1 = -5,\; c_1 = -2\)

\(a_2 = 6,\; b_2 = 2,\; c_2 = -7\)

Step 3: Apply Condition for No Solution

For no solution:

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

\[ \frac{k}{6} = \frac{-5}{2} \]

Step 4: Solve

\[ k = -15 \]

Step 5: Verify

\[ \frac{c_1}{c_2} = \frac{-2}{-7} = \frac{2}{7} \]

\[ \frac{-5}{2} \ne \frac{2}{7} \]

Condition satisfied ⇒ No solution ✔

Final Answer

\[ \text{The system has no solution when } k = -15. \]