Question

The sum of the digits of a certain two-digit number is 7. Reversing its digits increases the number by 9. What is the number?

Answer 1

b=4 a=3

`color(blue)("The first digit is 3 and the second 4 so the original number is 34")`

To be honest! It would be much quicker to solve by trial and error.

Explanation:

`color(magenta)("Building the equations")`

Let the first digit be `a`
Let the second digit be `b`

`color(blue)("The first condition")`

`a+b=7` ...............................(1)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("The Second condition")`

`color(green)("The first order value: ")`
`color(white)(xxxx)a` is a counting in tens. So actual value is `10xxa`
`color(white)(xxxx)b` is counting in units. So actual value is `1xxb`

`color(green)("The first Order Value" =10a+b)`...............................(2)
'-----------------------------------------------------------------------'
`color(purple)("The second order value:")`

`color(white)(xxxx)b` is a counting in tens. So actual value is `10xxb`
`color(white)(xxxx)a` is counting in units. So actual value is `1xxa`

`color(purple)("The second Order Value" =10b+a)`.........................(3)
'----------------------------------------------------------------------'

From the question
`color(red)("Equation (3)" - "Equation (2)"=9)`.................................(4)

`color(magenta)("|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||")`

`color(blue)("Putting it all together")`

`"Equation 4 becomes" ->(10b+a) -(10a+b)=9`

`9b-9a=9 color(white)(..)............................................(4_a)`
`a+b=7 color(white)(..).................................................(1)`

From equation (1)
`a=7-b`

Substitute in `(4_a)` giving:
`9b-9(7-b)=9`

`9b+9b-63=9`

`18b=72`

`color(blue)(b=72/18 = 4)`

Substitute in Equation (1) giving
`a+b=7-> a+4=7`

`color(blue)(a=3)`