Being able to think abstractly is a great skill to have. Learn how abstract equations can help you in refining your abstract-thinking ability. Also, learn what some of them look like and memorize their applications.

## Definition

An **abstract equation** is an equation that uses two or more variables. Neither variable can be solved unless the other variable is given. A simple example is y=x. Neither *y* nor *x* can be solved unless you know the other. If you knew for sure that x=1, then you would know *y*. But if you didn’t know that, then you wouldn’t be able to solve it. Usually, these types of equations are left as is. Because that is as far as you can go, they stay written like that.

You can have as many variables in an abstract equation as needed. Let me show you some examples of abstract equations.

See how all of the above have at least two variables? None can be solved unless you plugged in numbers for one or more of the variables. That is what sets abstract equations apart from other equations.

## Types of Abstract Equations

So, what types of abstract equations are there? There are actually many groups out there. The mathematicians of old and of today have done a really good job of categorizing the various types that exist out there. There are too many to list, so I will just show you some of the more common ones that you will encounter in school.

**1. Linear equations** These equations are the ones you see usually with a *y* and an *x*. They are called linear equations because when you start plugging in different values for *x* to get the *y* values and plot these points on a graph, the points will show a straight line.

**2. Quadratic equations** These look similar to the linear equations except there is an added x^2 somewhere in the equation. When you plot the points from this type of equation on a graph, you end up with a parabola.

**3. Sinusoidal equations** These equations involve the sine function and produce a repeating wave when graphed.

**4. Exponential equations** When you see the variable in the exponent area, you are looking at an exponential equation. When graphed, these produce a curve that starts out flat and then curves either up or down.

These are just a few of the many types of abstract equations there are out there. Each is useful for solving real-world problems. Let’s look at where these equations are used.

## Uses of Abstract Equations

It’s important to remember this: the whole world runs on math. If it weren’t for math, Earth wouldn’t spin around its axis or spin around the sun in such coordination as it does. Gravity wouldn’t work the way it does if it weren’t for math. Because of math, we expect things that go up to come down. Math is important! Here are just a few of the many areas in which abstract equations are used for our benefit.

**1. Physics** Forensic scientists use abstract equations to calculate whether a car had enough distance to stop safely or not at a crime scene. Roller coaster designers rely on abstract equations to calculate how steep or how curvy a coaster can safely be designed.

**2. Engineering** Airplane designers will use abstract equations to calculate whether the material they are using for the outside of the plane will withstand the extreme temperatures and pressure that the aircraft may be exposed to in flight. Car designers use abstract equations based on the materials they use to calculate how safe their cars are; how much force they can withstand without breaking.

**3. Signal processing** Abstract equations are used by cell phone designers to figure out a way to get the clearest signal to and from your cell phone. Abstract equations are also used to figure out a best way to keep the data traveling to and from your cell phone safely, securely, and privately.

Suffice it to say that the specific equations used in these cases are complex and involve many more variables than our basic linear equation.

## Why Learn It?

Why should you learn about abstract equations? First, because you will encounter these in your math schooling and it is important that you understand these. But more importantly, because the world around us involves abstract equations and knowing them will give you a better grasp of the mathematical phenomena that happen in the world around you.

## Lesson Summary

Abstract equations are simply equations with two or more variables. They cannot be solved unless data is provided. These equations are useful as mathematical models to tell you what you can and cannot do given a particular equation. You can graph these equations to show you the possibilities. Designers all over the world and in all areas use abstract equations to help solve their problems and create newer and better models.

## Abstract Equation Overview

Terms | Definitions |
---|---|

Abstract equation | an equation that uses two or more variables; neither variable can be solved unless the other variable is given |

Types | linear equations quadratic equations sinusoidal equations and exponential equations |

Uses | physics, engineering and signal processing |

## Learning Outcomes

Make an effort to accomplish these objectives when the lesson has been reviewed:

- Describe important characteristics of abstract equations
- Determine the types of abstract equations
- Detail the uses for these equations