# Understanding the Use of Linear Relation in Business

Updated September 24, 2022

Professionals use various methods to explain common concepts in mathematical equations. One equation that helps identify the direct relationship between two items is a linear association. Learning about linear associations can help you better understand this principle and the relationship between two variables. In this article, we discuss the definition of a linear relation, identify the linear equation components, and highlight several examples of common linear relationships.

## What is the definition of linear relation?

A linear relation or linear association is a term used in statistics to describe a link between two variables shown by a straight line. In graph form, the direction and speed of change in one variable directly affect the other variable in constant proportions. Linear associations are common throughout life. For example, the conversion of Fahrenheit to Celsius or the increase in the price of a product lowering its demand demonstrate common linear associations.

You can show linear associations on a graph or by using a mathematical equation. The simplicity of a linear association is in knowing that if your graph doesn't show a straight line, you've either graphed it incorrectly or the equation is not linear. The other critical element of linear associations is that the variable never equals more than two. Typically, the two variables used are x and y to represent the horizontal and vertical axis on a graph.

## What is the linear equation?

Using algebraic mathematics, a linear association is one that satisfies this equation:

y = mx + b

In this is equation, m represents the slope, and b represents where y intercepts the point on a graph, also called the constant of proportionality. The variables of m and b are constants and x is an independent variable. As a result, y is the dependent variable in this equation that relates directly to the other data. Here are some other considerations when using the linear equation:

### Linear relation variables

Within the linear relation formula, there are two variables, comprising x and y. In a mathematical formula, a variable describes a letter that represents a number whose value may change. It identifies a value you insert into the equation to complete the formula. In the example of the linear formula, x and y represent the variables.

The two types of variables are independent and dependent. An independent variable doesn't rely on other variables to determine its value. Within the linear equation, x represents the independent variable. The dependent variable is y, which relies on the value of x to determine its number. There are three basic principles of linear association variables:

• The independent variable changes the value of the dependent variable.

• The dependent variable has no ability to change the value of an independent variable.

• The value of the dependent variable relies on the value of the independent variable.

Related: How to Calculate Growth Rate (With Formulas and Examples)

### The constant of proportionality

Within the linear equation, a vital concept emerges, called the constant of proportionality. There is a direct relationship between two independent variables. By identifying this, you can determine the formula that describes one variable by its connection to another. For example, when you convert miles to kilometres, the constant of proportionality is 1.609. You can multiply the mile by 1.609 to determine the corresponding kilometre amount. Another example of the constant of proportionality is the relationship between mass and volume. This specific proportional constant is what science calls density. In this example, you write the mathematical formula as:

Mass = density x volume

Density is the constant linear association of a material's mass per unit volume. Suppose you were to plot the two independent variables of this equation, mass and volume, on a graph. You plot mass on the y-axis of the graph and volume on the x-axis. The straight line at the interception between the variables is the constant of proportionality or density.

Related: What is Quantitative Analysis?

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## Examples of linear associations

There are many ways that you can use linear equations to determine relationships in your professional work. Some examples of linear associations include:

### Budgeting to determine the cost of items

The simplest example of a linear association is when you're determining the cost of items. You likely use this linear equation without realizing its functionality throughout your profession. If your role involves any budgeting, you use this formula to determine the cost of items. For example, suppose you're purchasing new computers for work and you're preparing several budgets based on the number of units bought. You have already negotiated the best price possible and know that each computer costs \$500.

A linear equation in the most basic state shows that y represents the total amount of money for the new computers and is the dependent variable in this formula. The value of x represents the number of employees requiring new computers. The constant of proportionality is \$500, as the cost of the computer doesn't change and remains constant regardless of the other variables changing:

y = x(m) where:

• the total cost of computers = y

• the number of employees = x

• the constant portion = m

y = x(500)

For this example, suppose you needed to determine the cost of purchasing new computers for half the staff and all employees to complete your budget. Half the staff totals 12 people, with the full staff totally 24:

y = 12 x 500
y = \$6,000 for half the staff
y = 24 x 500
y = \$12,000 for the complete staff

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### Identifying earning potential

You can use a linear equation to identify potential earnings. For example, suppose you're a talent agent for a musical band. The venue you're working with is paying the band for an upcoming performance and you want to determine how much money the event can make for your client. You know the venue is paying \$500 for the performance plus an additional \$5 for every ticket sold. In this example, you can use a linear equation to identify the earning potential.

In this linear equation, y represents the total earnings and x represents the number of tickets sold. You also have the constant proportional amount of \$500 that the venue pays regardless of total ticket sales:

y = 5x + 500

By using this equation, you can input various values to figure out how many tickets to sell to reach the band's revenue goals. For example, if they sell 100 tickets:

y = (5 x 100) + 500
y = \$1000

Suppose your client has told you they want to make \$6,000 for that night's performance. They want to determine how many tickets to sell to meet this goal. You can use the same formula using the data you have. In this instance, you know that the variable of y is the total revenue goal of \$6,000 and that each ticket sold equates to \$5 for the band. You also know that the constant amount of \$500 paid by the venue organizer happens regardless of ticket sales, so your formula becomes:

y = 5x + 500

6,000 = 5(x) + 500
x = (6,000 - 500) / 5
x = 1,100 tickets sold to meet the band's revenue goal

### Determining variable costs

Another example of using linear equations is to determine variable costs. In this example, suppose you're a professional interior house painter and you have a customer wanting a quote. You charge a base price of \$100 per day for your service and \$10 for each square metre you paint. You can use a linear equation to determine your price quote based on the square metres the customer wants to be painted.

In this linear equation, y represents the total price and x represents the number of square metres:

y = 10x + 100

By using this equation, you can provide a customer with a quick sales quote. For example, your customer has a room that is 2.5 metres across by 3 metres deep. The square meterage of the room is 7.5 square metres, making the equation:

y = (10 x 7.5) + 100
y = \$175

Related: How to Calculate Variable Cost With Examples

### Conversion of imperial to metric

Another common example of a linear association is in the conversion of imperial measurements to the metric system. For example, when you convert feet to metres, pounds to kilograms, or Fahrenheit to Celsius, the variable of conversion is constant for the equation. To convert temperature, you can use the following linear equation where y represents the Celsius temperature and x represents the temperature in Fahrenheit:

y = 5/9 (x - 32)

Suppose the temperature you want to convert is 93°F:

y = 5/9 (93 - 32)
y = 33.9°C

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