This is the basic and conceptual question from linear equations in two variables in which we have given two equations 2x-3y=7, (a+b)x-(a+b-3)y=4a+b and we have to find the value of a and b for which it has infinite solutions

Kindly solve the above equations

RS Aggarwal, Class 10, chapter 3D, question no 23

## The equations are

2xâˆ’3yâˆ’7=0

(a+b)xâˆ’(a+bâˆ’3)yâˆ’(4a+b)=0

Here,Â a1â€‹=2,b1â€‹=âˆ’3,c1â€‹=âˆ’7

a2â€‹=a+b,b2â€‹=âˆ’(a+bâˆ’3),c2â€‹=âˆ’(4a+b)

The system ofÂ linear equations has infinite solutions

âˆ´â€‹a1â€‹â€‹/a2=b1â€‹â€‹/b2=c1/c2â€‹â€‹

â‡’2/(a+b)=-3/âˆ’(a+bâˆ’3)â€‹=-7/âˆ’(4a+b)â€‹

â‡’2/(a+b)=3/(a+bâˆ’3)â€‹

â‡’2a+2bâˆ’6=3a+3b

â‡’b=âˆ’aâˆ’6

Again, we have

â‡’3/(a+bâˆ’3)â€‹=7/(4a+b)â€‹

â‡’12a+3b=7a+7bâˆ’21

â‡’5aâˆ’4b=âˆ’21

PuttingÂ b=âˆ’aâˆ’6

â‡’5aâˆ’4(âˆ’aâˆ’6)=âˆ’21â‡’9a=âˆ’45

â‡’a=âˆ’5

Putting Â a=âˆ’5 in b=âˆ’aâˆ’6

â‡’b=âˆ’(âˆ’5)âˆ’6â‡’b=âˆ’1

â‡’a=âˆ’5,b=âˆ’1