# Systems of Linear Equations

### Topics

## Introduction to :

We know how to solve a system of two linear equations, not to mention what a system of equations actually looks like. We *also* know that a system of two linear equations has 0, 1, or infinitely many solutions. Now it's time to learn a bit more of the vocabulary that's used to describe the number of solutions to a system. While the phrases "a ton" and "more than its fair share" are moderately descriptive, there's more specific terminology we can use that will make your teachers happy, so let's dig in.

A system is **consistent** if it has at least one solution. It also won't be accused of waffling if it runs for political office. A system with no solutions is called **inconsistent**. It can only successfully run for office in Chicago.

Aww, don't look so sad, Chicagoans. It's funny because it's true.

### Sample Problem

Is the system of equations

consistent or inconsistent?

If we graph these, we get lines that intersect exactly once, at (4, 5):

Since these lines intersect, the system has a solution and is therefore consistent.

### Sample Problem

Is the system of equations

consistent or inconsistent?

The second equation is what we find if we multiply the first equation by 2, which means these two equations are actually the same line and every point on the line is a solution. Since this system has infinitely many solutions, it certainly has at least *one* solution. Infinity is more than 1.

You can trust us on this one...we've done the calculus. Anyway, the system is consistent.

If a linear system of two equations is consistent, or has at least one solution, there are two possibilities: either the system has exactly 1 solution, or it has infinitely many. Either one line crosses the other at some point, or both are the same line. Hm. Sounds like the plot of *Fight Club*.

We say a consistent linear system of two equations is **dependent** if the system has infinitely many solutions, and **independent** if the system has exactly 1 solution.

Here's how you can remember these terms: you only need to listen to the song "Miss Independent" a total of "1" time in order for it to be stuck in your head for the next week. In fact, it's probably already stuck there now. Ah, the power of suggestion.

A system of equations can be either **consistent** or **inconsistent**. If the system is consistent, it can be either **dependent** or **independent.**

Dr. Math has a nice way to think about why the words **consistent, inconsistent, dependent,** and **independent** make sense from an English language point of view. Be careful, however. While he does use the title "Dr., " we do not recommend that you let him perform a kidney transplant on you.

#### Example 1

Is the system of equations consistent or inconsistent? |

#### Example 2

Is the system of equations dependent or independent? |

#### Example 3

Is the system of equations dependent or independent? |

#### Exercise 1

Determine whether the following system of equations is consistent or inconsistent. If the system is consistent, determine whether it is dependent or independent.

#### Exercise 2

Determine whether the following system of equations is consistent or inconsistent. If the system is consistent, determine whether it is dependent or independent.

#### Exercise 3

Determine whether the following system of equations is consistent or inconsistent. If the system is consistent, determine whether it is dependent or independent.

#### Exercise 4

#### Exercise 5