Find the value of a and b if the following pair of linear equation has infinite number of solutions.
$3 x - (a + 1) y = 2 b - 1; 5 x + (1 - 2 a) y = 3 b$

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Solution

$3 x - (a - 1) y = 2 b - 1$ .................(1)
$5 x + (1 - 2 a) y = 3 b$ .................(2)
Here $a 1 = 3, b 1 = - (a + 1)$ and $c 1 = 2 b - 1$
Or $a 2 = 5, b 2 = (1 - 2 a)$ and $c 2 = 3 b$
Give the system of equations have infinite solutions
$\frac{a 1}{a 2} = \frac{b 1}{b 2} = \frac{c 1}{c 2}$
$\frac{3}{5} = \frac{- (a + 1)}{(1 - 2 a)}$
$\Rightarrow - 5 (a + 1) = 3 (1 - 2 a)$
$\Rightarrow - 5 a - 5 = 3 - 6 a$
$\Rightarrow a = 8$
$\frac{3}{5} = \frac{2 b - 1}{3 b}$
$\Rightarrow 9 b = 10 b - 5$
$\Rightarrow b = 5$

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