### What Is a Linear Relationship?

A linear relationship (or linear association) is a statistical term used to describe a straight-line relationship between a variable and a constant. Linear relationships can be expressed either in a graphical format where the variable and the constant are connected via a straight line or in a mathematical format where the independent variable is multiplied by the slope coefficient, added by a constant, which determines the dependent variable.

A linear relationship may be contrasted with a polynomial or non-linear (curved) relationship.

### Key Takeaways

- A linear relationship (or linear association) is a statistical term used to describe a straight-line relationship between a variable and a constant.
- Linear relationships can be expressed either in a graphical format or as a mathematical equation of the form y = mx + b.
- Linear relationships are fairly common in daily life.

### The Linear Equation Is:

Mathematically, a linear relationship is one that satisfies the equation:

$\begin{aligned} &y = mx + b \\ &\textbf{where:}\\ &m=\text{slope}\\ &b=\text{y-intercept}\\ \end{aligned}$

In this equation, “x” and “y” are two variables which are related by the parameters “m” and “b”. Graphically, y = mx + b plots in the x-y plane as a line with slope “m” and y-intercept “b.” The y-intercept “b” is simply the value of “y” when x=0. The slope “m” is calculated from any two individual points (x_{1}, y_{1}) and (x_{2}, y_{2}) as:

$m = \frac{(y_2 - y_1)}{(x_2 - x_1)}$

#### Linear Relationship

### What Does a Linear Relationship Tell You?

There are three sets of necessary criteria an equation has to meet in order to qualify as a linear one: an equation expressing a linear relationship can't consist of more than two variables, all of the variables in an equation must be to the first power, and the equation must graph as a straight line.

A linear function in mathematics is one that satisfies the properties of additivity and homogeneity. Linear functions also observe the superposition principle, which states that the net output of two or more inputs equals the sum of the outputs of the individual inputs. A commonly used linear relationship is a correlation, which describes how one variable changes in linear fashion to changes in another variable.

In econometrics, linear regression is an often used method of generating linear relationships to explain various phenomena. Not all relationships are linear, however. Some data describe relationships that are curved (such as polynomial relationships) while still other data cannot be parameterized.

### Linear Functions

Mathematically similar to a linear relationship is the concept of a linear function. In one variable, a linear function can be written as follows:

$\begin{aligned} &f(x) = mx + b \\ &\textbf{where:}\\ &m=\text{slope}\\ &b=\text{y-intercept}\\ \end{aligned}$

This is identical to the given formula for a linear relationship except that the symbol f(x) is used in place of *y.* This substitution is made to highlight the meaning that x is mapped to f(x), whereas the use of *y* simply indicates that x and y are two quantities, related by A and B.

In the study of linear algebra, the properties of linear functions are extensively studied and made rigorous. Given a scalar C and two vectors A and B from R^{N}, the most general definition of a linear function states that: $c \times f(A +B) = c \times f(A) + c \times f(B)$

### Examples of Linear Relationships

#### Example 1

Linear relationships are pretty common in daily life. Let's take the concept of speed for instance. The formula we use to calculate speed is as follows: the rate of speed is the distance traveled over time. If someone in a white 2007 Chrysler Town and Country minivan is traveling between Sacramento and Marysville in California, a 41.3 mile stretch on highway 99, and the complete the journey ends up taking 40 minutes, she will have been traveling just below 60 mph.

While there are more than two variables in this equation, it's still a linear equation because one of the variables will always be a constant (distance).

#### Example 2

A linear relationship can also be found in the equation distance = rate x time. Because distance is a positive number (in most cases), this linear relationship would be expressed on the top right quadrant of a graph with an X and Y axis.

If a bicycle made for two was traveling at a rate of 30 miles per hour for 20 hours, the rider will end up traveling 600 miles. Represented graphically with the distance on the Y axis and time on the X axis, a line tracking the distance over those 20 hours would travel straight out from the convergence of the X and Y axis.

#### Example 3

In order to convert Celsius to Fahrenheit, or Fahrenheit to Celsius, you would use the equations below. These equations express a linear relationship on a graph:

$\degree C = \frac{5}{9}(\degree F - 32)$

$\degree F = \frac{9}{5}(\degree C + 32)$

#### Example 4

Assume that the independent variable is the size of a house (as measured by square footage) which determines the market price of a home (the dependent variable) when it is multiplied by the slope coefficient of 207.65 and is then added to the constant term $10,500. If a home's square footage is 1,250 then the market value of the home is (1,250 x 207.65) + $10,500 = $270,062.50. Graphically, and mathematically, it appears as follows:

In this example, as the size of the house increases, the market value of the house increases in a linear fashion.

Some linear relationships between two objects can be called a "constant of proportionality." This relationship appears as

$\begin{aligned} &Y = k \times X \\ &\textbf{where:}\\ &k=\text{constant}\\ &Y, X=\text{proportional quantities}\\ \end{aligned}$

When analyzing behavioral data, there is rarely a perfect linear relationship between variables. However, trend-lines can be found in data that form a rough version of a linear relationship. For example, you could look at the sale of ice-cream and number of hospital visits as the two variables at play in a graph and find a linear relationship between the two.