# Linear Relationship Definition ## What is a linear relation?

A linear relationship (or linear association) is a statistical term used to describe a linear 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 variable dependent.

A linear relation can be contrasted with a polynomial or non-linear relation (curve).

### Key points to remember

• A linear relationship (or linear association) is a statistical term used to describe a linear relationship between a variable and a constant.
• Linear relationships can be expressed in graphical form or in the form of a mathematical equation of the form y = mx + b.
• Linear relationships are quite common in everyday life.

## The linear equation is:

Mathematically, a linear relation is that which satisfies the equation:

The

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

Thethere=mX+bor:m=slopeb=originally orderedTheThe

In this equation, “x” and “y” are two variables that are linked by the parameters “m” and “b”. Graphically, y = mx + b draws in the x-y plane a line with the slope “m” and the intercept “b”. The intercept “b” is simply the value of “y” when x = 0. The slope “m” is calculated from any two individual points (x1, y1) and (x2, y2) as:

The

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

m=(X2TheX1The)(there2Thethere1The)TheThe

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## What does a linear relationship tell you?

An equation must respect three sets of criteria necessary to be considered linear: an equation expressing a linear relation cannot include more than two variables, all the variables of an equation must be at the first power and the equation must represent graphically a straight line.

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

In econometrics, linear regression is a method often used to generate linear relationships to explain various phenomena. However, not all relationships are linear. Some data describe curved relationships (such as polynomial relationships) while other data cannot be set.

## Linear functions

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

The

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

TheF(X)=mX+bor:m=slopeb=originally orderedTheThe

This is identical to the formula given for a linear relation, except that the symbol f (x) is used in place of there. This substitution is made to highlight the sense that x is mapped to f (x), while the use of there simply indicates that x and y are two quantities, linked by A and B.

In the study of linear algebra, the properties of linear functions are widely studied and made rigorous. Given a scalar C and two vectors A and B of RNOT, the most general definition of a linear function indicates that: The

$c times f (A + B) = c times f (A) + c times f (B)$

vs×F(A+B)=vs×F(A)+vs×F(B)The

## Examples of linear relationships

### Example 1

Linear relationships are quite common in everyday life. Take the concept of speed, for example. The formula we use to calculate speed is as follows: the speed rate is the distance traveled over time. If someone in a 2007 Chrysler Town and Country white minivan travels between Sacramento and Marysville, California, a 41.3 mile stretch on Hwy 99, and the entire trip takes 40 minutes, they will have traveled just below from 60 mph.

Although there are more than two variables in this equation, it is 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 distance = rate x time equation. Since distance is a positive number (in most cases), this linear relationship would be expressed in the upper right quadrant of a graph with an X and Y axis.

If a bike made for two traveled at 30 miles per hour for 20 hours, the cyclist would end up traveling 600 miles. Represented graphically with the distance on the Y axis and time on the X axis, a line plotting the distance over these 20 hours would move directly from the convergence of the X and Y axes.

### 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:

The

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

°VS=95The(°F32)The

The

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

°F=59The(°VS+32)The

### Example 4

Suppose that the independent variable is the size of a house (measured in square feet) which determines the market price of a house (the dependent variable) when it is multiplied by the slope coefficient of 207.65 and is then added to the constant term $10,500. If the square footage of a house is 1,250, the market value of the house is (1,250 x 207.65) +$ 10,500 = \$ 270,062.50. Graphically and mathematically, it looks like this:

In this example, as the size of the house increases, the market value of the house increases linearly.

Certain linear relationships between two objects can be called “proportionality constants”. This relationship appears to be

The

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

TheYes=k×Xor:k=constantYes,X=proportional quantitiesTheThe

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