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:

\[ 4x + ky + 8 = 0, \qquad 2x + 2y + 2 = 0 \]

Solution

Step 1: Identify Coefficients

From the given equations,

\[ a_1 = 4, \quad b_1 = k \]

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

Step 2: 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 3: Apply the Condition

\[ \frac{4}{2} \neq \frac{k}{2} \]

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

\[ k \neq 4 \]

Conclusion

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

\[ \boxed{k = 4} \]

\[ \therefore \quad \text{The system has a unique solution for } k \neq 4. \]