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What Are Systems of Equations?

A blackboard with the equations 2x+y=5 and 1-y=5.

A system of equations is quite simply a collection of equations. Until now you have been working with one equation with one variable (one unknown). Now you’ll learn how to solve two equations with two variables (two unknowns).

In general, systems of equations can have as many unknowns as you want them to, but you need to have as many equations as unknowns to get a unique answer. In mathematics there in an entire field called linear algebra, which deals with solving systems of equations.

So why do you need systems of equations? The answer is simple: Often several things are dependent on each other, and then you need a tool that takes this into consideration. An example is buying children’s tickets and adults’ tickets at the same time. You know the total price of the tickets and how many tickets you bought, but what is the actual price of the two different types of tickets?

You’re going to learn three methods for solving systems of equations: Solving by graphing, the substitution method and the elimination method. All of these methods do exactly the same thing: They solve two equations with two variables. It doesn’t really matter which method you use while solving problems, but you are expected to know all three. Solving by graphing is the easiest, so let’s begin with that one. But before you start learning about the different methods there are some things you need to be aware of.


Important things about two equations with two variables

The solutions of linear systems of equations are made up of two values: an x value and a y value.
These two values are the first coordinate and the second coordinate of the intersection of the two graphs created by your equations.
In fact, the equations you are given can be rewritten into functions you can draw as graphs in a coordinate system.
Whether you use the substitution method or the elimination method, you will have two values in your solution. These two values can always be thought of as the coordinates of the intersection of the two graphs.
If you can’t find a solution, it means the graphs are parallel. In that case they have the same slope, but meet the second axis at different points. Because of that they will never intersect.
If you find an infinite number of solutions, it means that the graphs are identical. That means that they have the same slope, and meet the second axis at the same point. The graphs are on top of each other, and they intersect at every point.

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Graphic Representation of Equations