One of the most important and conceptual question from linear equations in two variables in which we have been asked to find the number a two digit number is such that the product of the digit is 35, when 18 is added to the number, the digit interchange their places.

Kindly give me a detailed solution of this question

RS Aggarwal, Class 10, chapter 3E, question no 16

Let the two digit no. be xy.

∴ Decimal expansion of xy=10×x+y

digits are interchanged, i.e. yx.

∴ Decimal expansion of yx=10×y+x

x×y=35{Given}

⇒x=35/y⟶(i)

Given that when 18 is added to the number, the digits are interchanged, i.e. yx.

∴(10x+y)+18=(10y+x)

⇒9x−9y+18=0

⇒9(x−y+2)=0

⇒x−y+2=0

⇒35/y −y+2=0{from eqn(i)}

⇒35−y²+2y=0

⇒y²−2y−35=0

⇒(y+5)(y−7)=0

∵y≠−5

∴y=7

Substituting the value of y in eqn(i), we get

x=35/7=5

Hence the two digit no. is 57.