Compare equations with one variable to equations with two variables. How are they alike? How are they differen

Question

what if an equation had three variables?

Answer 1

The more variables you have, the more difficult they are to solve them. Take a one variable equation, like x + 4 = 15. That's easy enough - you get one answer with the one equation. x can only be one value for this to make sense.

Now with a two variable equation, you don't have that singularity. Take y = 4x + 1. What does y equal? It depends on what x equals. If, say, x = 0, then y = 1. But if x = 1, then y = 5. So you don't have that singluar answer anymore - you have a set of answers which all would make the equation true. That's what a line represents - an set of x and y values that makes the equation true.

More variables makes for a more complicated solution set. If you have x + y + z = 1, then the value of x depends on the values of y and z, and those don't have to be related to each other at all. Now, you have a 3D shape that represents the solution set (as opposed to the 2D shape, or line, that represented the solution set to the two variable equation.

Finding a single solution, then, requires more than just one equation. For a two variable equation, you can get an exact answer if you have two equations. Same for a three variable equation - you need three different equations in order to get exact values for x, y, and z. When you have two or three equations that are meant to be used together to solve for your variables, you have a system of equations. For example, y = x + 3 and y = 2x + 5 are a system of equations that you can use to solve for a single x and y. When you do the math, you find that x = -2 and y = 1 - the system becomes true if and only if you use these two values for x and y. Graphically, this is represented by the intersection of the two lines - they meet at (-2, 1) when you graph both lines separately.