What is a 'Linear Relationship'
Linear relationship is a statistical term used to describe the relationship between a variable and a constant. Linear relationships can be expressed 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.
BREAKING DOWN 'Linear Relationship'
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 the superposition principle, which states that the net output of two or more inputs equals the sum of the outputs of the individual inputs.
Mathematically, a linear relationship is one that satisfies the equation:
y = mx + b
In this equation, “x” and “y” are two variables which are related by the parameters “a” and “b”. Graphically, y = mx + b plots in the xy plane as a line with slope “m” and yintercept “b”. The slope “m” is calculated from any two individual points (x_1, y_1) and (x_2, y_2) as
m = (y_2  y_1) / (x_2  x_1)
while the yintercept “b” is simply the value of “y” when x=0.
Mathematically similar to a linear relationship is the concept of a linear function. In one variable, a linear function can be written as
f(x) = mx + b
which 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*f(A +B) = c*f(A) + c*f(B)
Examples
Linear relationships are pretty common in daily life. The formula we use to calculate speed is 'rate of speed is distanced 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 they complete the journey ends up taking 40 minutes, they 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).
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, they'd end up raveling 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.
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.
Assume that the independent variable is the size of a house (as measured by square footage), 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 the home is $270,062.50. Graphically, and mathematically:
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 Y=KX, where k is the constant, and y and x are the proportional quantities.
When analyzing behavioral data, there is rarely a perfect linear relationship between variables. However, trendlines can be found in data that form a rough version of a linear relationship. For example you could look at the sale of icecream and number of drowning as the two variables at play in a graph and find a linear relationship between the two.

Nonlinear Regression
A form of regression analysis in which data is fit to a model ... 
Regression
A statistical measure that attempts to determine the strength ... 
Error Term
A variable in a statistical and/or mathematical model, which ... 
Algebraic Method
A mathematical means of solving a pair of linear equations. Algebraic ... 
Multiple Linear Regression  MLR
A statistical technique that uses several explanatory variables ... 
Linear Price Scale
A type of scale used on a chart that is plotted in such a way ...

Fundamental Analysis
Explaining Linear Relationships
A linear relationship describes the proportionality between an independent variable and a dependent variable. 
Professionals
Regression Analysis
CFA Level 1  Regression Analysis 
Professionals
Correlation and Regression
CFA Level 1  Correlation and Regression 
Active Trading
The Linear Regression Of Time and Price
This investment strategy can help investors be successful by identifying price trends while eliminating human bias. 
Economics
Understanding Regression
Regression is a statistical analysis that attempts to predict the effect of one or more variables on another variable. 
Active Trading
What's the Correlation Coefficient?
The correlation coefficient is a measure of how closely two variables move in relation to one another. If one variable goes up by a certain amount, the correlation coefficient indicates which ... 
Stock Analysis
The Top 5 DividendPaying Semiconductor Stocks for 2016 (CY, LLTC)
Discover the top 5 dividendpaying semiconductor stocks for 2016, with a summary, growth outlook and expected price target of each company. 
Professionals
Common Probability Distributions
CFA Level 1  Common Probability Distributions  Basics 
Stock Analysis
The Top 5 Large Cap Semiconductor Stocks for 2016 (AMAT, AVGO)
Discover the top five largecap semiconductors that are expected to grow in 2016, including a summary of each company and rational behind its expected growth. 
Economics
What Does a Relationship Manager Do?
A firm’s relationship manager works to maintain positive relationships with its customers and partner firms.

What are some of the more common types of regressions investors can use?
Learn about the most common types of regressions investors use to model asset prices including linear regressions and multiple ... Read Answer >> 
What is the difference between linear regression and multiple regression?
Learn the difference between linear regression and multiple regression and how multiple regression encompasses not only linear ... Read Answer >> 
What does it mean if the correlation coefficient is positive, negative, or zero?
Learn what the correlation coefficient between two variables is and what positive, negative and zero correlation coefficients ... Read Answer >> 
How can I run linear regressions in MATLAB?
Learn how to run linear regressions in MATLAB by loading data, specifying dependent and independent variables and using the ... Read Answer >> 
How can I use a regression to see the correlation between prices and interest rates?
Learn how to use linear regression to calculate the correlation between stock prices and interest rates by taking the square ... Read Answer >> 
Can you calculate more than two inputs with the production possibility frontier?
Understand how production possibility frontiers are used in business and learn more about how additional products may be ... Read Answer >>