Linear relationships describe a direct correlation between two variables, wherein a change in one variable results in a proportional change in the other, following a straight-line pattern on a graph.

Whenever you have a constant rate change, this gives a linear relationship.

Essentially, this is a relationship where one variable changes at a constant rate in relation to the other.

A linear relationship will always produce a straight line when graphed.

Since it has two variables, it has two dimensions which are represented by the x-axis and y-axis.

In this way, it compares the rate at which one variable changes against the other.

**How to Depict A Linear Relationship **

This can be done in two ways:

**A. As a straight line on a graph:**

This results in a line with a constant (fixed) slope.

It differs from other relationships (called polynomials or nonlinear relationships) which usually have curves with variable slopes.

**B. As a mathematical equation:**

For this, we use the general equation for all linear relationships:

y= mx + b

where

- y stands for the dependent variable y
- x is the independent variable
- m is a constant number which is the slope of the line
- b is the value (a constant) where the line cuts the y axis

**The Slope gives the Rate of change **

You can always know the rate of change of a linear relationship from the slope of its line.

In addition, the slope gives you other information like:

- The steeper your line, the greater the slope which means the higher the rate of change

- If the line is positively sloped, this means the two variables (x and y) are positively related. When x increases, y increases.

- If the line is negatively sloped, the variables are negatively related. When x increases, y decreases.

Let’s explore examples of linear relationships in real life:

**1. Constant speed **

If a car is moving at a constant speed, this produces a linear relationship.

For example, a car moving constantly at 50 km/ hour doesn’t change the rate at which it is moving. With each hour, its speed remains fixed

**2. Sales **

If you have a coffee shop, you can relate the sales you make with the number of sandwiches you sell.

This produces a linear relationship expressed in the following way:

Sales = price x sandwiches sold

y=2x which is a linear equation

where y = sales in dollars and x = sandwiches sold.

2 is the constant value that stands for the price of one sandwich

**3. Voltage in a Circuit**

Ohm’s law is a useful function that can be used to relate the voltage and current of a circuit.

It states that these two variables are directly proportional to each other.

This means if voltage doubles, current also doubles. This gives a linear relationship with a constant resistance as shown below (where resistance R is chosen as 200).

y=200x where y = voltage

x=current

200 is a constant value that stands for resistance. It is a property of a particular conductor.

**4. Celsius to Fahrenheit**

Linear relationships can help us convert from one unit of measurement to another.

For example, the formula that relates Fahrenheit and Celsius is:

C=5/9(F-32)

This can be expanded to:

C=5/9F – (5×32)/9.

The form of this equation conforms to the general linear equation y=mx +b that defines a linear relationship.

It can be written as

y=5/9x – 160/9

Where y is the temperature in Celsius

X is the temperature in Fahrenheit

-160/9 is the value of y where the line cuts the y axis

5/9 is the slope of the line

**5. Taxi Fare**

The fare that a taxi charges you can be shown to have a linear relationship with the miles covered for your journey.

Here is an example where the price is fifty dollars for every mile traveled.

y=50x

Where y is your fare and x is the miles

**6. Car Gas Mileage**

Gas mileage shows the distance that a specific car can cover on a given amount of gasoline.

This relationship can be modeled through a linear equation that relates gas mileage and your distance traveled.

**7. Distance Traveled **

If your speed is 40km per hour, we can show a linear relationship between the hours you travel and the distance you travel. This can be expressed as

y=40x

where y is your distance and x is the hours you have traveled

**8. Circumference of a circle **

The formula of the circumference of a circle is

C= 22/7(D)

This shows a linear relationship between the circumference and the diameter.

**Conclusion **

Our world is filled with linear relationships. In this section, we have seen how linear relationships richly add to our knowledge in mathematics, science, and everyday life.