Learn how beautifully simple linear relationships are and how easy they are to identify. Discover how you can see them in use in the world around you on an everyday basis and why they are useful. At the end of the lesson, test yourself with a quiz.

## What is a Linear Relationship?

As its name suggests, a **linear relationship** is any equation that, when graphed, gives you a straight line. Linear relationships are beautifully simple in this way; if you don’t get a straight line, you know you’ve either graphed it wrong or the equation is not a linear relationship. If you get a straight line and you’ve done everything correctly, you know it is a linear relationship.

## How to Identify Linear Relationships

There are only three criteria that equations must meet to qualify as a linear relationship. What are they? Let’s find out. To be called a linear relationship, the equation must meet the following three items:

1. The equation can have up to two variables, but it cannot have more than two variables.

2. All the variables in the equation are to the first power. None are squared or cubed or taken to any power. And also, none of the variables will be in the denominator. These are examples of equations that do not have a linear relationship.

You’ll notice that these equations have variables that are squared and cubed. One equation has a variable in the denominator. When graphed, none will yield a straight line.

3. The equation must graph as a straight line. Linear relationships such as *y* = 2 and *y* = *x* all graph out as straight lines. When graphing *y* = 2, you get a line going horizontally at the 2 mark on the y-axis. When graphing *y* = *x*, you get a diagonal line crossing the origin.

## Examples of Linear Relationships

There are equations in use in the real world today that meet all the criteria discussed above. Linear relationships are very common in our everyday life, even if we aren’t consciously aware of them. Take, for example, how fast things such as cars and trains can go. Have you ever thought about how their speeds are calculated? When a police officer gives someone a speeding ticket, how do they know for sure if the person was speeding? Well, they use a simple linear relationship called the **rate formula**.

This formula tells us that the speed of a certain object is calculated by dividing the distance traveled by the time it took to travel that distance. So, if someone spent 1 hour traveling a distance of 80 miles on a 55 mph road, then you can be sure that they were speeding because 80 miles divided by 1 hour gives you 80 mph. At first glance, this formula looks like it doesn’t fit the criteria because it looks like it has three variables. But, it really is a linear relationship because at least one of your variables will always be a constant depending on your problem. You can have a constant rate for which you have to solve for distance or time. The relationship would be 35 = *d* / *t* or whatever the given rate is. It’s the same if the distance is given as the constant, *r* = 100 / *t*.

Another example is that of converting temperature from Fahrenheit to Celsius. If you live in the United States, you probably use Fahrenheit, but if you discuss weather with a friend who lives in a different part of the world, you may need to convert the temperature to Celsius. You can use the **conversion formula** to convert one temperature type to the other:

You just saw two formulas: one for converting Fahrenheit to Celsius and the other for converting Celsius to Fahrenheit. This is the formula that is used when you use an automatic temperature converter app. And also, on classic mercury thermometers where it shows both Fahrenheit and Celsius together, you can check it by plugging various numbers into the equations to see if it matches.

You may also be familiar with linear relationships if you travel. When you visit another country, you will most likely need to exchange your currency for the currency of the place you are visiting. Let’s say, for example, you wanted to travel to Australia from the United States. As of May 2013, the exchange rate for converting Unites States dollars to Australian dollars is 1.0373. To figure out how much money you’d be getting back after the exchange, you would use this **exchange rate conversion formula**, which is a linear relationship:

The first equation is the general exchange rate conversion formula, and the second is the more specific one for converting United States dollars to Australian dollars.

Looking at all of these examples, you can see how they all meet the criteria for linear relationships. All the variables are to the first power; there are at most only two variables; and all graph to a straight line.

## How To Work with Linear Relationships

Working with linear relationships is very straightforward. It’s all about plugging in the right value for the right variable and making the calculations based on the formula. For example, let’s say I wanted to find out how fast I could run. I start to run, and I have to stop to catch my breath after 15 minutes. I’d only gone 1 mile. How fast was I going?

According to my formula, I need to plug in my distance and my time and then divide by two. So, I do that and I get .067 miles per minute. Not too bad for someone out of shape. Do you see how easy it is to work with linear relationships?

Looking back at our currency example, let’s say I wanted to exchange 500 United States dollars. How many Australian dollars would I end up with?

Looks like I would end up with 518.65 Australian dollars. Woohoo! I am $18.65 richer! Well, not really, but it feels like it. Can you see how I simply plugged in my values into the appropriate variable and made the appropriate calculation?

## Lesson Summary

There are only three criteria an equation must meet to qualify as a **linear relationship**:

- It can have up to two variables
- The variables must be to the first power and not in the denominator
- It must graph to a straight line

Working with linear relationships is straightforward and a matter of simply plugging in the right values into the right variables and making the appropriate calculations.

## About Linear Equations

- All linear equations, when graphed, form a line.
- A linear equation can have no more than two variables.
- The variable(s) in a linear equation cannot be located in the denominator.
- Linear equations are used in daily life.

## Learning Outcomes

Upon completing this lesson on linear relationships, ensure that you have the capacity to:

- Write the definition of a linear relationship
- List the criteria for a linear equation
- Apply linear relationships to the real world
- Use specific formulas and plug in the correct values to make calculations