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Pair of Linear Equations in Two Variables
CH03 Question Bank
CBSE_CH03_QUESTION_BANK
Grade 10

Question:

If $a_1b_2=a_2b_1$ but $a_1c_2
eq a_2c_1$, the pair of linear equations $a_1x+b_1y+c_1=0$, $a_2x+b_2y+c_2=0$ has:

A unique solution
No solution
Infinitely many solutions
Exactly two solutions

Step-by-Step Solution

Key Concept: $a_1b_2=a_2b_1$ is equivalent to $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}$, and $a_1c_2<br>eq a_2c_1$ is equivalent to $\dfrac{a_1}{a_2}<br>eq\dfrac{c_1}{c_2}$ — the no-solution condition.
$a_1b_2=a_2b_1 \Rightarrow \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}$, and $a_1c_2
eq a_2c_1 \Rightarrow \dfrac{a_1}{a_2}
eq\dfrac{c_1}{c_2}$. [0.5 Mark]

This is exactly the no-solution condition, so the system is inconsistent. [0.5 Mark]

Correct Answer: No solution
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