Find the values of a and b for which each of the following systems of linear equations has an infinite number of equations:

$2 x + 3 y = 7, (a + b + 1) x + (a + 2 b + 2) y = 4 (a + b) + 1.$

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Solution

$2 x + 3 y = 7, (a + b + 1) x + (a + 2 b + 2) y = 4 (a + b) + 1$

So for infinite solutions

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \frac{2}{a + b + 1} = \frac{3}{a + 2 b + 2} = \frac{7}{4 (a + b) + 1}$

$\frac{2}{a + b + 1} = \frac{3}{a + 2 b + 2} 3 a + 3 b + 3 = 2 a + 4 b + 4 a - b = 1 a = 1 + b - - - (1)$

$\frac{3}{a + 2 b + 2} = \frac{7}{4 (a + b) + 1} \frac{3}{a + 2 b + 2} = \frac{7}{4 a + 4 b + 1} 7 a + 14 b + 14 = 12 a + 12 b + 3 - 5 a + 2 b = - 11 5 a - 2 b = 11 P u t (1) h e r e, 5 (1 + b) - 2 b = 11 5 + 5 b - 2 b = 11 3 b = 11 - 5 3 b = 6 \Rightarrow b = 2 a n d a = 1 + 2 = 3$

a = 3 and b = 2

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Similar questions

2 x - 3 y = 7 (a + b) x - (a + b - 3) y = 4 a + b

2 x + 3 y = 7; (a + b + 1) x + (a + 2 b + 2) y = 4 (a + b) + 1

2 x + 3 y = 7 (a + b) x + (2 a - b) y = 21

a

b

2 x - 3 y = 7

(a + b) x - (a + b - 3) y = 4 a + b

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