Q. 103.7( 3 Votes )

# If 2x – 3y = 7 an

Answer :

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 :

a_{1}x + b_{1}y = c_{1} & a_{2}x + b_{2}y = c_{2} where

a_{1} & a_{2} are the coefficients of x

b_{1} & b_{2} are the coefficients of y

c_{1} & c_{2} 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__

………(i)

According to the problem:

a_{1} = 2

a_{2} = a + b

b_{1} = – 3

b_{2} = a + b – 3

c_{1} = 7

c_{2} = 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*5b + 5b = 6 ⇒ 30b = 6⇒ b =

### 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__

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