Systems of Linear Equations

Systems are graphs with more than one line on them. Three cases can occur in a system with two lines: 1. they can intersect at one point, 2. they don't touch each other (parallel lines), 3. they are the same line. One can solve a system of linear equations to figure out which case is valid for a given system. If they intersect the system yields one solution (the point of their intersection), otherwise infinite solutions for lines that are the same or no solution for parallel lines. There are multiple methods to solve these systems. The latter two have in common that they point they yield have to be plugged into the original equations to probe for the solution.

No Solution

When using the resulting point in both equations results in a false statement (e.g. $6 + 0 = 9$.

Infinite Solutions

When both equations result in constants equalling themselves (e.g. $-12 = -12$.

Graphing

One simply draws the lines on a graph and visually identifies the system's state.

Substitution

This method is ideal when one equation is solved for $x$ or $y$. The term that resolves to one variable in one equation can be plugged into the other equation to solve the system. Example:

$x - 3y = 11$, $y = 4x$ -> $x - 3(4x) = 11$ leads to $x = -1$, which can be plugged into the other original equation to solve for $y$: $y = 4(-1)$ -> $y = -4$.

Therefore the point $(-1,-4)$ is where both lines intersect.

For sets of line equations in standard form the substitution method can be set up by solving one of the equations for either $x$ or $y$. If this method is employed solved variables should be plugged into the original, unaltered equations.

The elimination method is recommended for equations that yield results heavy in fractions.

Elimination

This method requires both equations to be in standard form. The idea is to add both equations up so that one of the variables eliminates to $0$:
$3x - y = -5$, $-2x + y = 4$ -> $x = -1$

Like with the substitution method, the solved variable can be used to solve for the other by plugging it into one of the original equations.

If adding up the equations wouldn't eliminate one of the variables, simply multiply one equation by the number required to line it up with the other equation:
$3x - 2y = 8$
$4x + y = 7$ -> $\times 2$ results in the pair:
$3x - 2y = 8$, $8x + 2y = 14$

In some cases both equations may need to be multiplied by different numbers.

Linear Inequalities with Line Equations

The solution set to one of these is never a line. Instead they result in all the points above or below a line. Whether the line itself is included in the set depends on the sign used ($<, >$ vs $\leq, \geq$).
Since all points of the solution set are on the same side of the line, the solution can be probed by simply testing with any one point (e.g. $(0,0)$).

Systems of linear inequalities probe for overlapping areas of solution sets.