The sum of the digits of a two digit number is 12. If the new number formed by reversing the digits?

U + T = 12 The sum of the digits of a two digit number is 12.

10T + U + 54 = 10U + T Reversing the digits is greater than the original number by 54

You have two lines. They look like this: www.wolframalpha.com/input/?i=plot+U+%2B+T+%3D+12,+10T+%2B+U+%2B+54+%3D+10U+%2B+T
Where they cross, the values of x and y satisfy both equations simultaneously, so it is called a simultaneous system. There are two ways to find that point. Substitution is when you solve one equation for one variable and substitute that into the other equation.

U = 12 - T

10T + (12 - T) + 54 = 10(12 - T) + T

9T + 66 = 120 - 9T

18T = 54

T = 3, U = 9 <-- ANSWER

Elimination is when you add the equations in a way to eliminate one variable. Otherwise the process is the same.