Linear Relationship

A linear relationship describes a relation between two distinct variables – x and y in the form of a straight line on a graph. When presenting a linear relationship through an equation, the value of y is derived through the value of x, reflecting their correlation.

Linear relationships apply in day-to-day situations where one factor relies on another, such as an increase in the price of goods, lowering their demand. In any case, it considers only up to two variables to get an outcome.

• A linear relationship is one in which two variables have a direct connection, which means if the value of x is changed, y must also change in the same proportion.
• It is a statistical method to get a straight line or correlated values for two variables through a graph or mathematical formula.
• The number of variables considered in a linear equation never exceeds two.
• The correlation of two variables in day-to-day lives can be understood using this concept.

It best describes the relationship between two variables (independent and dependent) commonly represented by x and y. In the field of statistics, it is one of the most straightforward concepts to understand.

For a linear relationship, the variables must give a straight line on a graph every time the values of x and y are put together. With this method, it is possible to understand how variation between two factors can affect the result and how they relate to one another.

Let us take a real-world example of a grocery store, where its budget is independent variable and items to be stocked are the dependent variable. Consider the budget as $2,000, and the grocery items are 12 snack brands ($1-$2 per pack), 12 cold drink brands ($2-$4 per bottle), 5 cereal brands ($5-$7 per pack), and 40 personal care brands ($3-$30 per product). Because of budget constraints and varying prices, purchasing more of one will require purchasing lesser of the other.

Whether graphically or mathematically, y’s value is dependent on x, which gives a straight line on the graph. Here is a quick formula to understand the linear correlation between variables.

y = mx + b

In the formula, m denotes the slope. At the same time, b is the Y-intercept or the point on the graph crossing the y-axis with the x coordinate being zero. If the values of m, x, and b are given, one can easily get the value of y. One can graphically plot the same to show the linear relationship. Let us understand the process when the values for the x and y variables are assumed as follows in the sum below:

• x = 2, 4, 6, 8
• y = 7, 13, 19, 25

To calculate m, start by finding the difference pattern between the values of x and y and then put them as a fraction.

Hence, m = y2 – y1/x2 - x1

Putting the values from the x and y values in the above equation,

we get,

• m = 13-7/4-2
• m = 6/2
• m = 3

The next step is to find the hypothetical number (b) to be added or subtracted in the formula to get the value of y. As such,

y = mx + b

• y = 3*2 + 1
• y = 7.

Similarly, calculating the rest of the points, we get the following graph.

A linear relationship graph will look like this:

Let us take you through a detailed explanation of a linear equation or function.  When plotted on a graph, it will generate a straight line. A linear equation can occur in two forms – slope-intercept and standard form.

Slope-Intercept Form

It is one of the most recognizable linear functions in mathematics and is calculated on the x-y plane as follows:

y = mx + b

Here, m is the slope, b is the y-intercept, and x and y are two variables. Y-intercept occurs when the resultant line on the graph crosses the y-axis at a value. In this case, variable x must equal 0 at the point of the y-intercept.

Likewise, a slope represents how steep the line is and how to describe the relationship between the variables. For example, calculating two different points for two variables, i.e., x1, x2, and y1, y2, will provide the slope m.

m=(x2​−x1​) (y2​−y1​)​

Standard/General Form

It is another form of the linear function that is effective in understanding scenarios with two inputs (and no outputs) and can be derived as:

Ay + Bx = C

Again, x and y are two variables, whereas A, B, and C are constants in this equation. However, it is possible to arrive at the slope-intercept using the standard form.

For example, Ay + Bx = C

Ay = -Bx + C

Y = -Bx/A + C/A, which is essentially in the form of Y = mx + b

After putting the values in the above equation, one can make a linear graph using slope-intercept form.

Linear relationship examples are everywhere, such as converting Celsius to Fahrenheit, determining a budget, and calculating variable rates. Recently, a Bloomberg Economics study led by economists established a linear correlation between stringent lockdown measures and economic output across various countries. Moreover, they explained how moderate containment and mild social distancing could boost the economy.

A practical example of a linear equation could be cooking a homemade pizza. Here, two variables are the number of people to be served (constant or independent variable) and pizza ingredients (dependent variable). For example, suppose there is a pizza recipe for four, but only two people are there to consume it. To accommodate two people, cutting the number of ingredients to half would half the output.

Although linear and non-linear relationships describe the relations between two variables, both differ in their graphical representation and how variables are correlated.

Graphical Representation

A linear relationship will always produce a straight line on a graph to depict the relations between two variables. On the other hand, a non-linear relationship may create a curved line on the graph for the same purpose.

Change in Variables

In a linear relationship, a change in the independent variable will change the dependent variable. But this is not the case with a nonlinear relationship, for any changes in either variable will not affect the other.

Application Areas

A linear relationship best describes situations where variables are interdependent, such as exercise and weight loss. Here, exercising x times a day will significantly reduce a y amount of weight.

There is no linear association between variables in a nonlinear relationship, such as the effectiveness of a drug and dosage duration. It is because there could be several factors in between affecting the drug’s efficacy, such as –

• If the patient takes the medicines on time?
• Was it taken with the due procedure?
• Did the patient visit the doctor for the periodic check as suggested in the prescription?

Hence, the drug’s effectiveness determines by several factors, not just the dosage duration, which makes it a non-linear relationship. Many studies have been conducted to judge the viability of studying situations from the linear correlation perspective. This Harvard study has focused on some problem areas in this regard. It has also talked about how many situations are inevitably non-linear.

This comprehensive guide to the linear relationship discussed the equations, examples, and differences from the nonlinear relationship, along with key takeaways. To learn more about its use in finance, read the following articles –

• Regression
• Linear Regression in Excel
• Non-Linear Regression in Excel