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

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Given equation.......
$\begin{matrix} H e r e w e h a v e t w o l i n e a r e q u a t i o n : a_{1} x + b_{1} y + c_{1} = 0 - - - - - - - - - - (1) a_{2} x + b_{2} y + c_{2} = 0 - - - - - - - - - (2) N o w, c o n d i t i o n o f i n f i n i t e s o l u t i o n : \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} N o w, c o n s i d e r f r o m e q u a t i o n : F r o m e q u (1) & (2), 2 x + 3 y - 7 = 0 - - - - - - - - (i) (a - b) x + (a + b) y - (3 a + b - 2) - - - - - (i i) L e t s, t h e s e e q u^{n} c a m p a r e w i t h a b o v e t w o e q u^{n} : \Rightarrow 2 x + 3 y - 7 = 0 - - - - - - - - (i) a_{1} = 2, b_{1} = 3, c_{1} = - 7 \Rightarrow (a - b) x + (a + b) y - (3 a + b - 2) - - - - - (i i) a_{2} = (a - b), b_{2} = (a + b), c_{1} = - (3 a + b - 2) N o w, g i v e n c o n n d i t i o n : \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \frac{2}{a - b} = \frac{3}{a + b} = \frac{- 7}{- (3 a + b - 2)} f i n d i n g v a l u e o f a & b, \Rightarrow \frac{2}{a - b} = \frac{3}{a + b} = 2 a + 2 b = 3 a - 3 b ∴ a = 5 b - - - - - (i i i) A g a i n, \Rightarrow \frac{3}{a + b} = \frac{7}{(3 a + b - 2)} = 9 a + 3 b - 6 = 7 a + 7 b = 2 a - 4 b - 6 = 0 \Rightarrow a - 2 b - 3 = 0 \Rightarrow 5 b - 2 b - 3 = 0 \Rightarrow 3 b = 3, ∴ b = 1 p u t t h e v a l u e o f b i n e q u (i i i) a = 5 b, 5 \times 1 a = 5 & b = 1 N o w w e g e t v a l u e o f a = 5 & b = 1 I n t h i s c a s e t h e r e a r e i n f i n i t e c a s e s . \end{matrix}$

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Q. Question 2 (i)
For which values of ‘a’ and ‘b’ does the following pair of linear equations have an infinite number of solutions?
2x + 3y =7
(a - b)x + (a + b)y = 3a +b - 2

(i) For which values of a and b will the following pair of linear equations have an

infinite number of solutions?

(ii) For which value of k will the following pair of linear equations have no solution?

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

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

Q. Question 3
For what values of a and b will the following pair of linear equations have infinitely many solutions?
x + 2y = 1
(a - b)x + (a + b)y = a + b - 2

Question 2 (i)
For which values of ‘a’ and ‘b’ does the following pair of linear equations have an infinite number of solutions?
2x + 3y =7
(a - b)x + (a + b)y = 3a +b - 2

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

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