Question

If 2x – 3y = 7 and (a + b) x – (a + b – 3) y = 4a + b represent coincident lines, then a and b satisfy the equation

A. a + 5b = 0

B. 5a + b = 0

C. a – 5b = 0

D. 5a – b = 0

Answer 1

Given:

Equation 1: 2x – 3y = 7

Equation 2:(a + b)x + (a + b – 3)y = 4a + b

Both the equations are in the form of :

a₁x + b₁y = c₁ & a₂x + b₂y = c₂ where

a₁ & a₂ are the coefficients of x

b₁ & b₂ are the coefficients of y

c₁ & c₂ are the constants

When two sets of linear equations which are coincident then they will have infinite number of solutions since both the equations represent the same line

So we have to use the conditions for the infinitely many number of solution.

For the system of linear equations to have infinitely many solutions we must have

According to the problem:

a₁ = 2

a₂ = a + b

b₁ = – 3

b₂ = a + b – 3

c₁ = 7

c₂ = 4a + b

Putting the above values in equation (i) and solving the extreme left and middle portion of the equality

⇒ – 3(a + b) = 2(a + b – 3)

⇒ 5a + 5b = 6 …(ii)

Again We Solve for the extreme left and right side of the equality

⇒ 8a + 2b = 7a + 7b

⇒ a = 5b (iii)

We solve for a & b from

Equation (ii) & (iii)

We substitute the value of a from equation (iii) in equation (ii)

After substituting we get

5 x 5b + 5b = 6

⇒ 30b = 6

⇒ b = \(\frac{1}{5}\)

Putting the value of b in equation (iii) we get

a = 1

So

5b = a

The relationship between a and b for which the two equations represent coincident line is a = 5b or a – 5b = 0