For what values of  $a\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}b$ does the following pair of linear equations have an infinite numbers of solutions  $2x+3y=7,a(x+y)-b(x-y)=3a+b$

$2x+3y=7\cdots \cdots \left(1\right)$

$a(x+y)-b(x-y)=3a+b-2\cdots \cdots \left(2\right)$

Can be written as

$(a-b)x+(a+b)y=3a+b-2\cdots \cdots \left(3\right)$

$\left(1\right)\&\left(3\right)$ must satisfy $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$\begin{array}{l}\frac{a-b}{2}=\frac{a+b}{3}=\frac{3a+b-2}{7}\\ 3(a-b)=2(a+b)\\ 3a-3b=2a+2b\\ a=5b\cdots \cdots \cdots \left(1\right)\\ 7(a+b)=3(3a+b-2)\\ 7a+7b=9a+3b-6\\ 4b=2a-6\\ 3+2b=a\cdots \cdots \cdots \left(2\right)\end{array}$

From (1) and (2)

$5b=2b+3$

$3b=3\Rightarrow b=1$

$a=5$