Simultaneous Equations

I love Simultaneous Equations. It's so cool how you can take two completely random equations (2x + 8y = 20 and 2x + 4y = 12 for example) and just add or subtract them (2x + 8y = 16 - 2x + 4y = 12 comes out as 4y = 4, therefore y = 1 and plugging that in, 2x + 8 = 16, 8 + 8 = 16, 8/2 = 4, therefore, x = 4, so the answer is: (4, 1)) before you know it you have a pair of coordinates. Here's something really cool though. Say you have a pair of simultaneous equations like this: x + y = 15, y = x +3. That is telling you that y is the same number as x + 3. So, going back to x + y = 15, plug this in, it will be, x + x + 3 = 15, simplifying, 2x + 3 = 15, subtract 3, 2x = 12, divide, x = 6. Plug that in 6 + y = 15, 6 + 9 = 15. Therefore, the answer is: (6, 9). Here's a weird thing about simultaneous equations: say you have two equations, 5x + 3y = 35 and 5x + 3y = 33. Solving by subtraction you get 0 = 2. That can't be right. This type of simultaneous equation is called an inconsistent simultaneous equation. Another kind is: say you have x + y = 11 and 3x + 3y = 33, solving, we can multiply the first equation by 3 (another operation you can perform on equations) and then you have 3x + 3y = 33 and 3x + 3y = 33. Solving by subtraction we get, 0 = 0. Well thats not very helpful. These types of simultaneous equations are called equivalent equations (reheheheheheheheheheally?). That's all I have to say about simultaneous equations for right now. But here's one to work on. (See top right corner ;))