What is a system of equations?
Simply put, a system of equations consistent of at least two equations, and each equation in the system contains the same variables. Additionally, the variables must represent the same quantity. For example, if "x" in one equation represents "cost per person" but "number of people" in another equation, they cannot be part of the same system. This typically only requires consideration for problems in context. Out of context, if the variables are the same, we can consider them part of the same system. For example, here are two equations that make up a system:
Solutions to a System
A solution to a system of equations is a set of values for each variable in the system that satisfies all equations in the system. It is important to remember that the solution to one equation must satisfy every other equation in the system. If it doesn't satisfy all equations, it is not a solution to the system.
Let's look at a concrete example, using our system of two equations from above. I'll take some random value for x, plug it into one of the equations, and get a y-value. This represents a solution to that specific equation:
We can use this solution (3, 10) to plug in x = 3 and y = 10 into the other equation in our system:
As we can see, the point (3, 10) was a solution to one of the equations in the system, but it wasn't a solution to the other equation in the system. Since (3, 10) does not satisfy all equations in the system, it is not a solution to the system.
Solutions to systems of equations satisfy all equations in the system. Be careful for solutions that satisfy one, but not all equations. While a solution satisfies an equation, it does not necessarily represent a solution to the entire system.
Solutions to a System on a Graph
We know that solutions to a system of equations must satisfy all equations in the system. What does this look like on a graph? Let's start by examining the graph of a system of equations:
The point of intersection represents a single coordinate point that is a solution to both equations simultaneously. This intersection point is the point (2, 6). We can confirm algebraically that (2, 6) is a solution to all equations in the system:
two or more equations with the same variable, considered at the same time