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:
\[ x + 2y = 3, \qquad 5x + ky + 7 = 0 \]
Solution
Step 1: Write in Standard Form
\[ x + 2y – 3 = 0 \quad (1) \]
\[ 5x + ky + 7 = 0 \quad (2) \]
Step 2: Identify Coefficients
From equations (1) and (2),
\[ a_1 = 1, \quad b_1 = 2 \]
\[ a_2 = 5, \quad b_2 = k \]
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
\[ \frac{1}{5} \neq \frac{2}{k} \]
\[ k \neq 10 \]
Conclusion
The given system of equations has a unique solution for all real values of \(k\) except:
\[ \boxed{k = 10} \]
\[ \therefore \quad \text{The system has a unique solution for } k \neq 10. \]