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:

\[ 4x – 5y = k, \qquad 2x – 3y = 12 \]

Solution

Step 1: Write in Standard Form

\[ 4x – 5y – k = 0 \quad (1) \]

\[ 2x – 3y – 12 = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = 4, \quad b_1 = -5 \]

\[ a_2 = 2, \quad b_2 = -3 \]

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

\[ \frac{4}{2} = 2, \qquad \frac{-5}{-3} = \frac{5}{3} \]

\[ 2 \neq \frac{5}{3} \]

Hence, the condition for a unique solution is satisfied for all values of \(k\).

Conclusion

The given system of equations has a unique solution for:

\[ \boxed{\text{All real values of } k} \]

\[ \therefore \quad \text{The system has a unique solution for every real value of } k. \]