I know that this is basic, but I don't get it. Say we have the system,
[itex]x + y = 32[/itex]
[itex]3x + 2y = 70[/itex]
Each equation represents a graph, and the solution is the point where the two lines intersect.

Here is where I am confused:
[itex]x + y = 32[/itex]
[itex]2x + 2y = 64[/itex]
What is the proof that the two equations are equivalent / represent the same graph?

I read that two equations are equivalent if they share the same implication:

Your first two equations are not equivalent, since their graphs are different. They do intersect at a point, though. Assuming that they intersect at (2, 3), the equations x = 2 and y = 3 would be equivalent to the first pair of equations you have above. (Note that I just made up a solution, so my equations x = 2 and y = 3 aren't actually equivalent to yours.)

For your second system of equations, both equations represent exactly the same line. Each solution of the first equation is also a solution of the second equation. In different words, if (x_{0}, y_{0}) is a solution (i.e., makes the equation a true statement) of the first equation, it will also be a solution of the second equation. For this example, the two equations are equivalent.

1) If the two equations have one solution (which is one pair (x_{i}, y_{i})) then I treat x and y in the equations as two (different) values.
[itex]x+y=32[/itex]
[itex]3x+2y=70[/itex]

2) (x+y=32) is equivalent to (2x+2y=64), since they share the same set of coordinates, therefore the same graph. Lets say set A has the coordinates of (x+y=32), and set B of (3x+2y=70). The solution is the set, [itex]A \cap B[/itex].

Two equations are equivalent if they have exactly the same solution set. If (x, y) is in the solution set of the first equation then x+ y= 32. Multiplying both sides of the equation by 2, 2x+ 2y= 64 so (x, y) is in the solution set for that equation. That proves that the solution set for x+ y= 32 is a subset of the solution set for 2x+ 2y= 64. Now we have to do it the other way: if (x, y) is in the solution set for 2x+ 2y= 64, then they satisfy that equation. Dividing both sides by 2, x+ y= 32 so (x, y) is also in the solution set for that equaton and the solution set for 2x+ 2y= 64 is a subset of the solution set for x+ y= 32. Since each is a subset of the other, they are equal and the two equations are equivalent.