Finding Value of k for Unique Solution

Video Explanation

Question

Find the value of \(k\) for which the system of equations \(kx – y = 2\) and \(6x – 2y = 3\) has a unique solution.

Solution

Step 1: Write in Standard Form

\[ kx – y – 2 = 0 \]

\[ 6x – 2y – 3 = 0 \]

Step 2: Identify Coefficients

For equation (1): \(a_1 = k,\; b_1 = -1\)

For equation (2): \(a_2 = 6,\; b_2 = -2\)

Step 3: Apply Condition for Unique Solution

A pair of linear equations has a unique solution if:

\[ \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \]

\[ \frac{k}{6} \ne \frac{-1}{-2} \]

\[ \frac{k}{6} \ne \frac{1}{2} \]

Step 4: Solve

\[ k \ne 3 \]

Final Answer

\[ \text{The system has a unique solution for all values of } k \ne 3. \]

Verification Insight

If \(k = 3\), then:

\[ \frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{-1}{-2} = \frac{1}{2} \]

Ratios become equal ⇒ No unique solution ❌