System of linear equations: Unique, infinite and no solutions

Solving a system of two linear equations should be familiar to most of you. There are several methods of solving it, including substitution and subtraction of equations from each other. However, sometimes there can be three equations with three unknowns or even two equations with three unknowns. It is important to identify when those equations have unique, infinite amount or no solutions at all. The easiest way to do it is to solve the simultaneous equations. It is possible to think of a system of linear equations geometrically, where the solution is at the intersection of lines or planes. Thus the intersection can be a point, a line or a plane. Here is how to identify the amount of solutions that the system of equations has:

Unique solution there is only one set of variables that satisfy all equations. Intersection is a point.

No solutions no set of variables satisfy all equations, usually you get 1 = 0 when solving the system. No intersection of all equations in one point.

Infinite amount of solutions infinite amount of variables satisfy the equation, meaning at least one free variable. Intersection in a line or plane.

Example: Solve the system of linear equations:

First rewrite (1.1) as y = 2 − 3x to substitute into (1.2):

(1.1)3 + y = 2 ⇒ y = −1

The answer is: (1,−1). It can also be represented graphically, as an intersection of two lines in a single point.

Example: Solve the system of linear equations:

Rewrite equation (1.4) as z = −1 − 4x to substitute into (1.3):

There are not enough equations to find a unique solution, so we can do a substitution.

Let x =λ.

So our solution is:

Which means that there are infinite amount of solutions. Graphically it can be represented as two planes meeting in a line:

Example: Solve the system of linear equations:

Which is not true, thus there are no solutions to this system of linear equations. It can be seen as three planes that do not intersect in the same point: