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. Unlike a nonlinear relationship, which can produce curves, bends, or other complex shapes on a graph, a linear relationship will always appear as a straight line. For example, if you earn $15 for every hour you work, then doubling your hours doubles your pay, and tripling your hours triples your pay. The connection is simple, steady, and predictable as every increase in one variable leads to the same proportional increase (or decrease) in the other.
Examples of Linear Relationships in Real Life
1. Speed and Distance (Constant Speed)
If you’re driving at a steady speed of 60 km/h, then in 1 hour you’ll travel 60 km, in 2 hours 120 km, and in 3 hours 180 km. Distance increases at the same rate as time, as long as the speed is constant. On a graph, time and distance form a straight line, making this a linear relationship.
2. Sales
If you own a coffee shop and sell sandwiches for $2 each. The total sales you make are directly tied to how many sandwiches you sell. For every extra sandwich sold, your sales increase by $2. This creates a linear relationship that can be written as:
Sales = Price × Number of Sandwiches
Or in equation form: y = 2x,
where y represents total sales in dollars, x is the number of sandwiches sold, and 2 is the constant price of each sandwich.
3. Voltage in a Circuit
Ohm’s Law tells us that voltage and current are directly proportional in a circuit, as long as resistance remains constant. In other words, if the current doubles, the voltage doubles too. For example, with a resistance of 200 ohms, the relationship is expressed as:
y = 200x
where y is the voltage, x is the current, and 200 is the resistance of the conductor. This direct proportionality makes it a classic example of a linear relationship.
4. Celsius to Fahrenheit
Converting between Celsius and Fahrenheit is another example of a linear relationship. The formula is:
C = (5/9)(F – 32)
which can be expanded to:
C = (5/9)F – 160/9
This fits the standard linear form y = mx + b, where y is temperature in Celsius, x is temperature in Fahrenheit, 5/9 is the slope, and –160/9 is the point where the line crosses the y-axis. The straight-line relationship makes it easy to move between the two scales.
5. Taxi Fare
Taxi fares often follow a simple linear pattern. If a cab charges $50 for every mile traveled, then the total fare increases steadily with distance. This can be written as:
y = 50x
where y is the fare in dollars and x is the miles traveled. Each additional mile adds the same fixed cost, making the relationship perfectly linear.
6. Car Gas Mileage
Gas mileage shows how far a car can travel on a set amount of fuel, and it can also be expressed linearly. For example, if your car runs 30 miles per gallon, then the total distance traveled grows in direct proportion to the gallons of fuel you use. Each extra gallon adds the same distance, forming a straight-line relationship between fuel and distance.
7. Distance Traveled
Speed is another simple linear relationship. If you’re driving at a constant speed of 40 km/h, the distance you cover increases steadily with every passing hour. Mathematically, this looks like:
y = 40x
where y is the distance traveled and x is the time in hours. Every extra hour adds 40 km to your total.
8. Circumference of a circle
The circumference of a circle depends directly on its diameter, which makes it a linear relationship. The formula is:
C = (22/7)D
Here, C is the circumference and D is the diameter. The fraction 22/7 (an approximation of π) is the constant multiplier, meaning that as the diameter increases, the circumference grows proportionally in a straight-line manner.
9. Distance Walked and Steps Taken
If your stride is about 0.8 meters per step, then 10 steps move you 8 meters, 100 steps move you 80 meters, and 1,000 steps move you 800 meters. The distance increases at the same rate with each step. Because every step adds the same amount of distance, this is another real-life example of a linear relationship.
10. Water Flowing Into a Tank at a Constant Rate
Picture filling a tank with water at a steady flow of 5 liters per minute. After 1 minute you’ll have 5 liters, after 10 minutes 50 liters, and after 20 minutes 100 liters. The amount of water increases evenly with time, so plotting time against water volume produces a straight line.
11. Wages and Hours Worked
Imagine you’re working at a part-time job where you’re paid $15 an hour. If you work 1 hour, you earn $15; 2 hours gets you $30; 5 hours gets you $75. The amount of money you make increases at the same rate for every extra hour you work, so if you plot hours against pay, the graph is a straight line. This is a perfect example of a linear relationship.
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
And we have a calculator that can help you find the constant and variables in one click. Check it out here: Linear Relationship Calculator
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:
If the line is negatively sloped, the variables are negatively related. When x increases, y decreases.
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.