Question

If 2x - 3y = 7 and (a+b) x - (a+b-3) y =4a +b have an infinite number of solutions then​

Answer 1

Answer:

Given: Two equations, 2x – 3y = 7

⇒ 2x – 3y – 7 = 0

(a + b) x – (a + b – 3) y = 4a + b

(a + b) x – (a + b – 3) y – (4a + b) = 0

We know that the general form for a pair of linear equations in 2 variables x and y is a1x + b1y + c1 = 0

and a2x + b2y + c2 = 0.

Comparing with above equations,

we have a1 = 2,

b1 = - 3,

c1 = - 7;

a2 = a + b,

b2 = - (a + b – 3),

c2 = - (4a + b)

Since, it is given that the equations have infinite number of solutions, then lines are coincident and

So,

Let us consider

Then, by cross multiplication, 2(a + b – 3) = 3(a + b)

⇒ 2a + 2b – 6 = 3a + 3b

⇒ a + b + 6 = 0 … (1)

Now consider

Then, 3(4a + b) = 7(a + b – 3)

⇒ 12a + 3b = 7a + 7b – 21

⇒ 5a – 4b + 21 = 0 … (2)

Solving equations (1) and (2),

5 × (1), (5a + 5b + 30) – (5a – 4b + 21) = 0

⇒ 9b + 9 = 0

⇒ 9b = - 9

⇒ b = - 1

Substitute b value in (1),

a - 1 + 6 = 0

a + 5 = 0

a = - 5

∴ a = - 5; b = - 1

Answer 2

Answer:

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