Pair of Linear Equations in Two Variables
NCERT Exemplar
CBSE
Grade 10
Question:
If the system $2x + 3y = 7$ and $(a + b)x + (2a - b)y = 21$ has infinitely many solutions, then:
(a) $a = 1, b = 5$
(b) $a = 5, b = 1$
(c) $a = -1, b = 5$
(d) $a = 5, b = -1$
Step-by-Step Solution
Key Concept: Infinitely many solutions $\Rightarrow \dfrac{2}{a+b} = \dfrac{3}{2a-b} = \dfrac{7}{21} = \dfrac{1}{3}$.
From $\dfrac{2}{a+b} = \dfrac{1}{3} \Rightarrow a + b = 6$. [0.5 Mark]
From $\dfrac{3}{2a-b} = \dfrac{1}{3} \Rightarrow 2a - b = 9$.
Adding equations: $3a = 15 \Rightarrow a = 5$, so $b = 6 - 5 = 1$. [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Setting up equal ratios = 1/3: 0.5 Mark
Solving simultaneous equations for $a=5, b=1$: 0.5 Mark
Correct Answer: $a = 5, b = 1$
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