Question

The sum of the digits of a two-digit number is 12. The number with the digits reversed is 15 times the original tens' digit. Find the original number.
I don't need the answer, just the equations! Thanks!

Answer 1

$a-digit\ of\ tens\\b-digit\ of\ ones\\10a+b-the\ number\\10b+a-the\ number\ with\ reversed\ digits\\\\\left\{\begin{array}{ccc}a+b=12&\to b=12-a\\10b+a=15(10a+b)\end{array}\right\\\\\text{Substitute}\\\\10(12-a)+a=15(10a+12-a)\\10(12-a)+a=15(9a+12)\qquad\text{use distributive property}\\(10)(12)+(10)(-a)+a=(15)(9a)+(15)(12)\\120-10a+a=135a+180\\120-9a=135a+180\qquad\text{subtract 120 from both sides}\\-9a=135a+60\qquad\text{subtract 135a from both sides}\\-144a=60\qquad\text{divide both sides by (-144)}\\\\a=-\dfrac{60}{144}$

a is negative and it's not a digit.

Conclusion: Such a number does not exist.