Find the values of $a$ and $b$ for which the pair of equations $2x+3y=7$ and $(a-b)x+(a+b)y=3a+b-2$ has infinitely many solutions.
Step-by-Step Solution
Key Concept: Set up the equal-ratios condition across all three coefficient pairs, and solve the resulting simultaneous equations in $a,b$.
For infinitely many solutions: $\dfrac{2}{a-b}=\dfrac{3}{a+b}=\dfrac{7}{3a+b-2}$. [1.0 Mark]
From the first two: $2(a+b)=3(a-b)\Rightarrow 2a+2b=3a-3b\Rightarrow a=5b$ — (1). [1.0 Mark]
From the first and third: $2(3a+b-2)=7(a-b)\Rightarrow 6a+2b-4=7a-7b\Rightarrow a=9b-4$ — (2). [1.5 Marks]
Equating (1) and (2): $5b=9b-4\Rightarrow 4b=4\Rightarrow b=1$; then $a=5(1)=5$. [1.0 Mark]
Verification: $a-b=4,\ a+b=6,\ 3a+b-2=14$; ratios $\dfrac24=\dfrac36=\dfrac{7}{14}=\dfrac12$ — all equal, confirming $a=5,\ b=1$. [0.5 Mark]
Correct Answer: