Finding Value of k for Inconsistent System

Video Explanation

Question

Find the value of \(k\) for which the system of equations \(3x + y = 1\) and \((2k – 1)x + (k – 1)y = 2k + 1\) is inconsistent.

Solution

Step 1: Write in Standard Form

\[ 3x + y – 1 = 0 \]

\[ (2k – 1)x + (k – 1)y – (2k + 1) = 0 \]

Step 2: Identify Coefficients

\(a_1 = 3,\; b_1 = 1,\; c_1 = -1\)

\(a_2 = 2k – 1,\; b_2 = k – 1,\; c_2 = -(2k + 1)\)

Step 3: Apply Condition for Inconsistent System

For inconsistency:

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

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

Step 4: Solve

\[ 3(k – 1) = 2k – 1 \]

\[ 3k – 3 = 2k – 1 \]

\[ k = 2 \]

Step 5: Verify

\[ \frac{c_1}{c_2} = \frac{-1}{-(2k+1)} = \frac{1}{2k+1} \]

For \(k = 2\):

\[ \frac{1}{2k+1} = \frac{1}{5} \neq \frac{3}{3} = 1 \]

Condition satisfied ⇒ Inconsistent ✔

Final Answer

\[ \text{The system is inconsistent when } k = 2. \]