Consistency of a Pair of Linear Equations
Video Explanation
Question
For which value(s) of \( \lambda \), do the pair of linear equations
\[ \lambda x + y = \lambda^2, \qquad x + \lambda y = 1 \]
have (i) no solution (ii) infinitely many solutions (iii) a unique solution?
Solution
Step 1: Write in Standard Form
\[ \lambda x + y – \lambda^2 = 0 \quad (1) \]
\[ x + \lambda y – 1 = 0 \quad (2) \]
Step 2: Identify Coefficients
From equations (1) and (2),
\[ a_1 = \lambda, \quad b_1 = 1, \quad c_1 = -\lambda^2 \]
\[ a_2 = 1, \quad b_2 = \lambda, \quad c_2 = -1 \]
(i) Condition for No Solution
A pair of linear equations has no solution if
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]
\[ \frac{\lambda}{1} = \frac{1}{\lambda} \Rightarrow \lambda^2 = 1 \Rightarrow \lambda = \pm 1 \]
Check the third ratio:
\[ \frac{c_1}{c_2} = \lambda^2 \]
For \( \lambda = -1 \):
\[ \frac{a_1}{a_2} = -1, \quad \frac{b_1}{b_2} = -1, \quad \frac{c_1}{c_2} = 1 \]
Hence, the system is inconsistent.
\[ \boxed{\lambda = -1} \]
(ii) Condition for Infinitely Many Solutions
A pair of linear equations has infinitely many solutions if
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
From above, \( \lambda = 1 \).
Check:
\[ \frac{a_1}{a_2} = 1, \quad \frac{b_1}{b_2} = 1, \quad \frac{c_1}{c_2} = 1 \]
Hence, the equations represent the same line.
\[ \boxed{\lambda = 1} \]
(iii) Condition for a Unique Solution
A pair of linear equations has a unique solution if
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
\[ \lambda \neq \frac{1}{\lambda} \Rightarrow \lambda^2 \neq 1 \]
\[ \boxed{\lambda \neq \pm 1} \]
Conclusion
(i) No solution for:
\[ \boxed{\lambda = -1} \]
(ii) Infinitely many solutions for:
\[ \boxed{\lambda = 1} \]
(iii) Unique solution for:
\[ \boxed{\lambda \neq \pm 1} \]