The sum of two numbers is 1 and their difference is 9 what are the two numbers?

x + y = 1
x - y = 9

You have two lines. They look like this:
www.wolframalpha.com/input/?i=plot+x+%2B+y+%3D+1,+x+-+y+%3D+9

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.

x + y = 1
y = 1 - x Substitute this for y in the other equation.

x - (1 - x) = 9
2x - 1 = 9
2x = 10
x = 5 Put this into either original equation to find y.

5 - y = 9
-y = 4
y = -4

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

x + y = 1
x - y = 9
------------ ADD
2x = 10
x = 5 <-- SAME ANSWER