SOLUTION: please help me: the sum of the digits of a two-digit number is 12. if the digits are reversed, the new number is 15 more than twice the original number. what is the original num

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Question 158041: please help me:
the sum of the digits of a two-digit number is 12. if the digits are reversed, the new number is 15 more than twice the original number. what is the original number?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let t=tens digit and u=units digit


ANY two digit number can be written in the form of: tens_digit*10+units_digit


Algebraically this looks like: n=10t%2Bu where "n" is any two digit number


If you reverse the digits of "n" you get m=10u%2Bt where "m" is the reversed number (ie if n=27 then m=72)


Since the "sum of the digits of a two-digit number is 12", this means that t%2Bu=12


Also, since the "new number is 15 more than twice the original number", this tells us that 10u%2Bt=2%2810t%2Bu%29%2B15


10u%2Bt=20t%2B2u%2B15 Distribute


10u%2Bt-20t-2u=15 Subtract 20t from both sides. Subtract 2u from both sides


8u-19t=15 Combine like terms.


-19t%2B8u=15 Rearrange the terms.


So we have the system:


system%28t%2Bu=12%2C-19t%2B8u=15%29



19%28t%2Bu%29=19%2812%29 Multiply the both sides of the first equation by 19.


19t%2B19u=228 Distribute and multiply.


So we have the new system of equations:
system%2819t%2B19u=228%2C-19t%2B8u=15%29


Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:


%2819t%2B19u%29%2B%28-19t%2B8u%29=%28228%29%2B%2815%29


%2819t%2B-19t%29%2B%2819u%2B8u%29=228%2B15 Group like terms.


0t%2B27u=243 Combine like terms. Notice how the x terms cancel out.


27u=243 Simplify.


u=%28243%29%2F%2827%29 Divide both sides by 27 to isolate u.


u=9 Reduce.


------------------------------------------------------------------


19t%2B19u=228 Now go back to the first equation.


19t%2B19%289%29=228 Plug in u=9.


19t%2B171=228 Multiply.


19t=228-171 Subtract 171 from both sides.


19t=57 Combine like terms on the right side.


t=%2857%29%2F%2819%29 Divide both sides by 19 to isolate t.


t=3 Reduce.


So our answer is t=3 and u=9.


This means that the original number is n=10%283%29%2B9=39 and the new number is m=10%289%29%2B3=93