Question

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

ANS a = 5 and b = 1

Answer 1

Solution is in the attachment...

Answer 2

Given Equation is 2x + 3y = 7

= > 2x + 3y - 7 = 0   

On comparing with a1x + b1y + c1 = 0, we get

a1 = 2, b1 = 3, c1 = -7


Given Equation is a(x + y) - b(x - y) = 3a + b - 2  

= > a(x + y) - b(x - y) - (3a + b - 2) = 0

= > ax + ay - bx + by - (3a + b - 2) = 0

= > x(a - b) + y(a + b) - (3a + b - 2) = 0

On comparing with a2x + b2y + c2 = 0, we get

a2 = (a - b), b2 = (a + b), c2 = -(3a + b - 2).


Given that the equation has infinite number of solutions.

= > (a1/a2) = (b1/b2) = (c1/c2)

= > (2/a - b) = (3/a + b) = (7/3a + b - 2)

Now,

(1)

= > (a1/a2) = (c1/c2)

= > (2/a - b) = (7/3a + b - 2)

= > 6a + 2b - 4 = 7a - 7b

= > a - 9b = -4   --------- (1)



(2)

= > (a1/a2) = (b1/b2)

= > (2/a - b) = (3/a + b)

= > 2a + 2b = 3a - 3b

= > a - 5b = 0  

= > a = 5b  ------- (2)


Substitute (2) in (1), we get

= > a - 9b = -4

= > 5b - 9b = -4

= > -4b = -4

= > b = 1.


Substitute b = 1 in (1), we get

= > a - 9b = -4

= > a - 9 = -4

= > a = 5.




Therefore the values of a = 5 and b = 1.



Hope this helps!