Finding Value of k for No Solution

Video Explanation

Question

Find the value of \(k\) for which the system of equations \(x + 2y = 5\) and \(3x + ky + 15 = 0\) has no solution.

Solution

Step 1: Write in Standard Form

\[ x + 2y – 5 = 0 \]

\[ 3x + ky + 15 = 0 \]

Step 2: Identify Coefficients

\(a_1 = 1,\; b_1 = 2,\; c_1 = -5\)

\(a_2 = 3,\; b_2 = k,\; c_2 = 15\)

Step 3: Apply Condition for No Solution

For no solution:

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

\[ \frac{1}{3} = \frac{2}{k} \]

Step 4: Solve

\[ k = 6 \]

Step 5: Verify

\[ \frac{c_1}{c_2} = \frac{-5}{15} = -\frac{1}{3} \]

\[ \frac{1}{3} \ne -\frac{1}{3} \]

Condition satisfied ⇒ No solution ✔

Final Answer

\[ \text{The system has no solution when } k = 6. \]