# 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. As an example,

$\begin{aligned} x+2y & =2 \\ -x+y & =1 \end{aligned}$

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.

A system of linear equations can have any number of variables and any number of equations. For a given system, we could have one solution, no solutions or infinitely many solutions.

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## Solutions

In a system of linear equations, each equation typically represents some kind of constraint on the problem. The **solution** to a system of equations will assign a value to each variable in the system, and these values will satisfy every equation in the system. An equation is **satisfied** if the assigned values make the equation a true statement.

$\begin{aligned} 3x+y &= 5 \\ x-y &= -1 \end{aligned}$

Is $x=1,y=2$ a solution to the system above?

Assign the given values to the equations to see if they satisfy the equations:$\begin{aligned} 3x+y &= 5 \\ 3(1)+(2) &\stackrel{?}{=} 5 \\ 5 &= 5 \\ \\ x-y &= -1 \\ (1)-(2) &\stackrel{?}{=} -1 \\ -1 &= -1 \end{aligned}$

Both equations are satisfied, so $x = 1,y=2$ is a solution.

Keep in mind that a solution must satisfy every equation in the system in order to be a solution.

$\begin{aligned} 3x+y &= 5 \\ x-y &= -1 \end{aligned}$

Is $x=0,y=5$ a solution to the system above?

Assign the given values to the equations to see if they satisfy the equations:$\begin{aligned} 3x+y &= 5 \\ 3(0)+(5) &\stackrel{?}{=} 5 \\ 5 &= 5 \\ \\ x-y &= -1 \\ (0)-(5) &\stackrel{?}{=} -1 \\ -5 &\not= -1 \end{aligned}$

Although the first equation is satisfied, the second equation is not. This means that $x =0,y=5$ is not a solution.

It is possible that there are infinitely many solutions for a system.

[Example of a given system that has infinitely many solutions]]

It is also possible that a system of equations has no solutions.

[Example of a given system that has no solutions]]

[2 example problems about identifying solutions]

## Methods for Solving

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)

**Cite as:**System of Linear Equations (Reference Card).

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/andy-test-1/