Linear Systems: Why Does Linear Combination Work (Graphically)?

A system of linear equations consists of multiple linear equations. You can think of this as multiple lines graphed on one coordinate plane. Three situations can arise when looking at such a graph. Either:
1) No point(s) are shared by all lines shown
2) There is one point that all lines cross through
3) The lines lie on top of one another, so there are many points that the lines have in common.

There are four commonly used tools for solving linear systems:
– graphing
– substitution
– linear combination
– matrices
Each has its own advantages and disadvantages in various situations, however I often used to wonder about why the linear combination approach works. My earlier post explains why it works from an algebraic perspective. This post will try to explain why it works from a graphical perspective.

Consider the linear system:

\begin{cases}y=-3x+2\\y=x-6\end{cases}

which, when graphed, looks like:

Continue reading Linear Systems: Why Does Linear Combination Work (Graphically)?

Solving Systems Of Linear Equations

What is a “system” of linear equations?

A “system of linear equations” means two or more linear equations that must all be true at the same time.

When represented symbolically, a system of equations will usually have some sort of grouping symbol to one side of them, such as the curly brace below, which is intended to convey that the set of equations should be considered all at once. For example:

\begin{cases}y=-3x+2\\y=x-6\end{cases}

When graphed, all of the equations in a system will be shown on the same set of axes, so that they can be compared to one another easily:

What is “a solution” to a linear system?

A solution to a system of linear equations is Continue reading Solving Systems Of Linear Equations