1. Systems of Equations

So far in this course, we’ve met equations like \(y = 2x - 4\) . We’ve seen that such equations can have many solutions. For example, both \((x, y) = (0, -4)\) and \((x, y) = (2, 0)\) are solutions to this equation. We often visualize the full set of solutions to the equation in the form of a graph:

Now we want to think about balancing multiple equations at the same time.

1.1. Linear Systems of Two Variables

When we have multiple constraints that we need to satisfy simultaneously, we typically write them with a curly brace, like this:

\[\begin{split}\begin{cases} \ \ \ x + y = 4\\ -x + y = 0\\ \end{cases}\end{split}\]

This is called a system of equations.

system of equations

A system of equations* is a set of equations (usually grouped with a curly brace) that we want to simultaneously balance

solution

A solution to a system of equations is an assignment of values for the variables (\(x = ..., y = ...\)) that will solve all of the equations in the system.

Note

The solution to the system here is \((x, y) = (2, 2)\). We can check a solution by verifying that all of the equations in the system are balanced.