For what values of $a$ and $b$ does the following pair of linear equations have an infinite number of solutions?
$2x + 3y = 7$ and $(a - b)x + (a + b)y = 3a + b - 2$.
Step-by-Step Solution
Key Concept: $\\dfrac{2}{a-b} = \\dfrac{3}{a+b} = \\dfrac{7}{3a+b-2}$. From 1st pair: $2a + 2b = 3a - 3b \Rightarrow a = 5b$. From 2nd pair: $3(3a+b-2) = 7(a+b) \Rightarrow 9a + 3b - 6 = 7a + 7b \Rightarrow 2a - 4b = 6$. Substitute $a = 5b \Rightarrow 10b - 4b = 6 \Rightarrow 6b = 6 \Rightarrow b = 1, a = 5$.
$\\dfrac{2}{a-b} = \\dfrac{3}{a+b} \Rightarrow 2a + 2b = 3a - 3b \Rightarrow a = 5b$. [1.0 Mark]
$\\dfrac{3}{a+b} = \\dfrac{7}{3a+b-2} \Rightarrow 9a + 3b - 6 = 7a + 7b \Rightarrow 2a - 4b = 6$. [1.0 Mark]
Substitute $a = 5b \Rightarrow 2(5b) - 4b = 6 \Rightarrow 6b = 6 \Rightarrow b = 1, a = 5$. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Deriving $a = 5b$: 1.0 Mark
Forming second equation $2a - 4b = 6$: 1.0 Mark
Solving $a = 5, b = 1$: 1.0 Mark
Correct Answer: