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Question

For what values of a and b does the following pair of linear equations have infinite number of solution ?
$$2x + 3y = 7, a(x + y ) - b(x - y) = 3a + b -2$$


Solution

$$2x+3y=7$$
$$ax-bx+ay+by=3a+b-2$$
(or) $$2x+3y=7$$ …………$$(1)$$
$$(a-b)x+(a+b)y=3a+b-2$$ ………….$$(2)$$
Given: the pair of linear equations have infinite number of solutions
$$\dfrac{2}{a-b}=\dfrac{3}{a-b}=\dfrac{7}{3a+b-2}$$
$$\Rightarrow \dfrac{2}{a-b}=\dfrac{3}{a+b}$$
$$\dfrac{3}{a+b}=\dfrac{7}{3a+b-2}$$
$$\Rightarrow 2a+2b=3a-3b$$
$$\Rightarrow -a=-5b$$
$$\therefore a=5b$$
$$3(3a+b-2)=7(a+b)$$
$$9a+3b-6=7a+7b$$
$$2a-4b-6=0$$
$$a-2b=3$$
Put $$a=5b$$ in $$a-2b=3$$ we get
$$5b-2b=3$$
$$\Rightarrow 3b=3$$
or $$b=1$$
$$\therefore a=5\times 1=5$$, $$b=1$$
Hence $$a=5, b=1$$.

1257506_1327231_ans_1fabf598b5314922be822593c425f5ae.PNG

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