Systems of linear equations.

A group of two (or more) (linear) equations together. A system of equations will have either 0, 1, or (an infinite) solutions (i.e. points that are solutions to both (or all) the equations).

1:
If two (or more) lines intersect, their linear equations are both (or all) satisfied by the intersection point, i.e. it's a solution to both (or all) equations. In a system of two (or more) linear equations, this common single intersection point solves the system. It is the 1 solution case, called an independent system. In a system of two linear equations, if the slopes of the equations' lines are different, (y=m1x+b and y=m2x+b, the y-intercepts b are irrelevant, they can different or the same) the lines will interesect and the system is independent.
Example:
Here is a system of two linear equations and their corresponding slope-intercept forms and their lines. The slopes differ, thus they intersect at one point which is the one point that solves both equations. The solution point can be determined by the substitution method or the addition method. In this random example, it's (56/17,-9/17), approximately (3.29,-0.53).

0:
If two lines are parallel, their system of equations has no solution (the 0 solution case, called an inconsistent system). In a system of two linear equations, if the slopes of their lines are the same (and their y-intercepts are different, i.e.y=mx+b1 and y=mx+b2), the lines are parallel and do not have any intersection and the system is inconsistent.
Example:
Here is a system of two linear equations and their corresponding slope-intercept forms and their lines. The slopes are the same (and the y-intercepts different), thus the lines are parallel and they have no intersection point, so there are no solutions for this system.

:
The ∞ solutions case, called dependent, is sort of a combo of the other two cases. If the two equations are actually the same just expressed differently, they have the same slope and the same y-intercept, i.e. they have the same line. The lines are identical, on top of each other. Every point on them is a solution to both equations.
Example:
Here is a system of two linear equations and their corresponding slope-intercept forms and their lines. The slopes are the same and the y-intercepts are the same, thus the lines are coincident and they intersect at every point on the line, so there are ∞ number of solutions for this system.

Systems of linear inequalities