Systems of linear equations (also known as simultaneous equations) are a set of equations that share common unknowns. Typically, we aim to manipulate this set of equations to extract the values of the variables. For example:
$$ 2x + 3y = 13\\ x + 4y = 14 $$
By writing one variable in terms of the other, we can use substitution to solve for one and then the other:
$$ x = 14 - 4y $$
$$ 2(14-4y) + 3y = 13\\ -5y + 28 = 13\\ 5y = 15\\ y = 3\\ x = 14-4(3) = 2 $$
And we get that $x=2, y=3$.
For a system of linear equations, there are three possibilities: one answer (most commonly), no answers or infinite solutions.
The case of no solutions occurs when the system of equations is inconsistent. This means that the system is essentially self-contradicting, for example:
$$ x + y = 3\\ x + y = 4 $$
There is no possible way for both of these to be true at the same time.
Infinite solutions occur when equations are multiples of one another. For example:
$$ x + y = 4\\ 3x + 3y = 12 $$
Notice that the second equation simplifies to the first, and the system is effectively only one equation - $x+y=4$. Without further restrictions, there are infinite pairs $(x,y)$ that satisfy the equation.
There is a powerful geometric interpretation of these solutions. The solution to a system of linear equations is the intersection between the two lines they draw. In the example case:
Why? Well, the line $2x+3y=13$ is essentially the set of all points $(x,y)$ that satisfy that equation, and the same is true when we draw $x+4y=14$. Finding the intersection of the lines yields the point $(2,3)$, which by definition satisfies both of these equations. As such, a solution is the intersection of these lines.
Now if we plot the inconsistent system of equations, we see that there are no intersections: