Question 130266
There are actually two ways to solve this problem.  Either way, your first equation is correct.  {{{x+y=7}}}.


The first way is to derive two additional equations.


If x is the 10s digit and y is the ones digit then 10x plus y equals the number.


{{{10x+y=n}}}


Now, if we reverse the digits, y becomes the 10s digit and x becomes the ones digit and the number is increased by 9, so:


{{{x + 10y=n+9}}}


Add -9 to both sides of this last equation to get {{{x+10y-9=n}}}.  Now we have two things that equal n so we can set these two expressions equal to each other:


{{{10x+y=x+10y-9}}}


{{{9x-9y=-9}}}


{{{x-y=-1}}}


Add this last equation to your very first equation ({{{x+y=7}}}) term by term:


{{{2x+0y=6}}}


{{{x=3}}}


From {{{3+y=7}}} we get {{{y=4}}}, therefore the number is 34.


The second way to solve the problem is to realize that the difference between any two-digit number and the result of reversing the digits of that two-digit number is a multiple of nine, and that the multiplier of 9 is the difference between the two digits (for example 25 and 52 differ by 27, 2 and 5 differ by 3 and 27 is 3 times 9).  Since reversing the digits in this problem resulted in a number that was 9 larger, the 10s digit had to be 1 smaller than the ones digit.  This way you could have written {{{x-y=-1}}} directly and solved from there.