System of Linear Equations (Reference Card)

A system of linear equations is a collection of linear equations. The equations are all to be considered at the same time. Systems of equations are used to solve problems in which there are a couple of constraints, and there are certain numerical values that satisfy those constraints.

As an example,

\[ \begin{align} x+2y & =2 \\ -x+y & =1 \end{align} \]

is a system of equations that has two variables \(x\) and \(y.\) The solution to a linear system is an assignment of numbers to the variables that satisfy every equation in the system. In the example above, there is one solution: \(x = 0, y=1.\) When the equations are graphed, the lines intersect at the solution.

The graph of the system of equations written above.

There are several different methods for solving systems of linear equations. They are linked below.

• Substitution Method
• Elimination Method
• Graphing Method (link)
• Matrices Method (link)