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Question
Seven times a two-digit number is equal to four times the number obtained by reversing the digits. If the difference between the digits is 3. Find the number.
Solution
Let the digits at units and tens place of the given number be x and y respectively. Thus, the number is `10y +x`.
The difference between the two digits of the number is 3. Thus, we have `x-y=+-3`
After interchanging the digits, the number becomes `10x+y`.
Seven times the number is equal to four times the number obtained by reversing the order of the digits. Thus, we have
` 7 (10y+x)=4(10x +y)`
` ⇒ 70 y + 7x = 40x +4y`
`⇒ 40x +4y -70y -7x =0`
` ⇒ 33x - 66y=0`
` ⇒ 33(x-2y)=0`
` ⇒ x - 2y =0`
So, we have two systems of simultaneous equations
` x-y =3`
` x- 2y =0`
`x- y = -3`
` x- 2y =0`
Here x and y are unknowns. We have to solve the above systems of equations for xand y.
(i) First, we solve the system
`x - y = 3`
` x- 2y =0`
Multiplying the first equation by 2 and then subtracting from the second equation, we have
`(x-2y)-2(x-y)=0-2xx3`
`⇒ x - 2y -2x +2y =-6`
` ⇒ -x =-6`
` ⇒ x=6`
Substituting the value of x in the first equation, we have
` 6 -y =3`
`⇒ y = 6-3 `
`⇒ y = 3`
Hence, the number is `10xx3+6 =36`
(ii) Now, we solve the system
`x -y =-3`
` x - 2y =0`
Multiplying the first equation by 2 and then subtracting from the second equation, we have
`( x - 2y)- 2( x -y)=0 - (-3xx2)`
`⇒ x - 2y -2x +2y =6 `
` ⇒ -x =6`
` ⇒ x =-6`
Substituting the value of x in the first equation, we have
` -6 -y =-3`
`⇒ y = -6 +3`
` ⇒ y = -3`
But, the digits of the number can’t be negative. Hence, the second case must be removed.
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