SOLUTION: 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? I know that the answer is 34, and I know that one

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: 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? I know that the answer is 34, and I know that one      Log On


   

Question 130266: 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?
I know that the answer is 34, and I know that one of the equations is x+y=7, but I do not know the other part to the system of equations. Thanks for helping me!!!
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
There are actually two ways to solve this problem. Either way, your first equation is correct. x%2By=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%2By=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+%2B+10y=n%2B9

Add -9 to both sides of this last equation to get x%2B10y-9=n. Now we have two things that equal n so we can set these two expressions equal to each other:

10x%2By=x%2B10y-9

9x-9y=-9

x-y=-1

Add this last equation to your very first equation (x%2By=7) term by term:

2x%2B0y=6

x=3

From 3%2By=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.