### Systems of linear equations: Systems of linear equations

### The notion of a system of linear equations

The notion of linear equation will now be extended to systems of linear equations.

General terminology A** system of linear equations **is understood to be one or more linear equations with one or more unknowns.

A **solution** of the system of equations is a list of values of the unknowns that, when substituted into each equation of the system, gives a set of equations that are valid (a true statement). Not every system of equations has a solution: the system \[ \left\{\;\begin{aligned} x + y\;&= 1 \\ x+ y\;&=2\end{aligned} \right.\] for example, has no solution. Such a system is called **inconsistent.**

**Solving** a system of equations is the determination of all the solutions. The result is also called the **solution.**

A system of linear equations is often solved by **reduction**, that is, by replacing it via elementary operations by another **system of** **linear equations** that is both simpler than the previous one, and the same solution. We speak of an **elementary reduction** if all steps in the reduction process consists of elementary operations. **Elementary operations** are multiplication of an equation by a nonzero number, addition of two equations, and interchange of two equations.

Two systems of linear equations are called **equivalent** if they have the same solution because they can be converted into each other via elementary reduction.

Below, we discuss how a system of linear equations can be seen as a system of vector equations.

Vector form The system of equations \[\left\{\;\begin{aligned} 2x + 3y\;&= 1 \\ \phantom{2}x+\phantom{3}y\;&=1\end{aligned} \right.\] can also be written in **vector form** as \[x\cdot\!\!\cv{2\cr 1\cr}+y\cdot\!\!\cv{3\cr 1\cr}= \cv{1\cr 1\cr}\] In this form, solving the system is nothing but finding an expression of the vector \(\cv{1\cr 1\cr}\) as a linear combination of the vectors \(\cv{2\cr 1\cr}\) and \(\cv{3\cr 1\cr}\).

We will often see that a linear algebra problem eventually leads to solving a system of linear equations. Therefore we pay considerable attention to finding solutions.

Number of solutions

A system of linear equations can have more equations than unknowns. We give an example of two equations with one unknown (a dynamic example at the bottom of this page). You can see that in this case there are three possibilities:

- There is no solution: the equations may be
**inconsistent**. That means that no solution of one equation form the system is also a solution of the other. Also, it may be that one of the equations has no solution. - There is exactly one solution.
- There is more than one solution: in this case the number of solutions is even infinite, because all the values of the unknown will satisfy both equations.