Condition for Unique Solution of a Pair of Linear Equations

Video Explanation

Question

Find the value of \(k\) for which the following system of equations has a unique solution:

\[ kx + 2y = 5, \qquad 3x + y = 1 \]

Solution

Step 1: Write in Standard Form

\[ kx + 2y – 5 = 0 \quad (1) \]

\[ 3x + y – 1 = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = k, \quad b_1 = 2 \]

\[ a_2 = 3, \quad b_2 = 1 \]

Step 3: Condition for Unique Solution

A pair of linear equations has a unique solution if

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

Step 4: Apply the Condition

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

\[ k \neq 6 \]

Conclusion

The given system of equations has a unique solution for all real values of \(k\) except:

\[ \boxed{k = 6} \]

\[ \therefore \quad \text{The system has a unique solution for } k \neq 6. \]