Applications of Linear Graphs 

Linear graphs are a powerful tool in mathematics and everyday life, used to represent relationships between two variables that change at a constant rate. These graphs are essential in various fields, from economics to physics, helping us understand trends and predict outcomes. In this blog, we'll explore the definition, properties, and real-life applications of linear graphs with practical examples that highlight their importance.

1.0What is a Linear Graph?

A linear graph is a visual representation of a linear equation, typically in the form y = mx + b, where:

• m is the slope of the line, indicating the rate of change between the two variables.
• b is the y-intercept, which is the point where the graph intersects the y-axis.

In simpler terms, a linear graph shows how one variable changes in relation to another in a straight-line pattern. The consistency in the slope means that the relationship between the two variables is constant, making linear graphs ideal for modeling proportional relationships.

2.0Properties of Linear Graphs

Linear graphs have several important properties that make them easily recognizable and highly useful:

1. Straight Line: As the name suggests, the graph of a linear equation is always a straight line. The slope m dictates whether the line is steep or shallow, while the y-intercept b determines its position on the graph.
2. Constant Slope: The rate of change between the two variables remains the same across the entire graph. This means that for any two points on the graph, the change in y-values divided by the change in x-values is constant.
3. No Curvature: Unlike quadratic or exponential graphs, linear graphs do not curve. The relationship between the variables is consistent.
4. Symmetry: A linear graph does not exhibit any type of curvature or bending, making it a simple and predictable tool for analysis.

3.0Practical Applications of Linear Graphs

Linear graphs have numerous applications in everyday life. Let’s take a look at some of the examples of application of linear graphs in real life:

1. Business and Economics

One of the most common applications of linear graphs is in business and economics, particularly in analyzing costs and revenues. For instance, if you run a business that has a fixed cost and a variable cost (like production costs), you can represent the total cost as a linear equation.

• Example: Suppose a company has a fixed monthly cost of $500 (like rent, utilities, etc.) and a variable cost of $10 for every unit produced. The total cost C for producing x units can be expressed as C(x) = 10x + 500, where x is the number of units.

A linear graph would represent this equation, where the x-axis shows the number of units and the y-axis shows the total cost. The slope of the graph (10) indicates the increase in cost per unit, while the y-intercept (500) shows the fixed cost.

2. Distance-Time Relationships

In physics, linear graphs are used to represent relationships between different physical quantities. A great example is the relationship between distance and time when an object is moving at a constant speed. The equation y = mx + b is commonly used to represent this relationship, where y is distance, x is time, and m is the speed of the object.

• Example: If a car travels at a constant speed of 60 km/h, the equation representing the distance traveled over time is d(t) = 60t, where t is time in hours, and d(t) is distance in kilometers.

The graph of this equation will be a straight line with a slope of 60, meaning that for each hour, the car covers 60 kilometers. This graph can be useful for determining how long it will take the car to reach a certain distance or how far it will travel in a given time.

3. Finance: Loan Repayment

Another practical application of linear graphs is in the context of loan repayment. In this case, the total amount paid over time can often be represented by a linear equation, especially in scenarios with fixed monthly payments.

• Example: If you take out a loan of $5,000 at an interest rate of 5% per year, and you agree to repay it in monthly installments of $200, the total amount paid over time can be modeled by the equation y = 200x + 5000, where x is the number of months and y is the total amount paid.

A linear graph can be used to visually track how the loan balance decreases over time, and the slope of the line would represent the rate of repayment.

4. Temperature Over Time

Linear graphs can also be used in meteorology to model the change in temperature over time under certain constant conditions. For instance, if a city experiences a constant temperature change each day (say, it increases by 2°C every hour), this can be graphed as a linear function.

• Example: The temperature starts at 20°C at 8 AM and increases by 2°C every hour. The temperature at t hours can be represented by the equation T(t) = 2t + 20, where t is the time in hours and T(t) is the temperature in Celsius.

This linear relationship allows us to predict the temperature at any given hour, helping meteorologists plan weather forecasts.

5. Speed and Fuel Consumption

In transportation, linear graphs are used to represent the relationship between speed and fuel consumption in vehicles. If a car uses fuel at a constant rate while traveling at a constant speed, this relationship can be graphed as a linear equation.

• Example: Suppose a car consumes fuel at a rate of 0.1 liters per kilometer. The equation for fuel consumption f(x) after traveling xx kilometers is f(x) = 0.1x, where x is the number of kilometers driven.

This allows drivers to estimate how much fuel will be used for a trip, which is useful for budgeting and route planning.

4.0Solved Examples on Linear Graphs 

Example 1: Graph the equation y = 2x + 1.

Solution:
The equation y = 2x + 1 is in the form y = mx + b, where:

• m = 2 is the slope of the line.
• b = 1 is the y-intercept (the point where the line crosses the y-axis).

To graph this, we can follow these steps:

1. Plot the y-intercept: Start by plotting the point where x = 0, which is (0, 1).
2. Use the slope: The slope m = 2 means that for every 1 unit increase in x, y increases by 2 units. Starting from (0, 1), move 1 unit to the right (increase x by 1), and 2 units up (increase y by 2). This gives the point (1, 3).
3. Draw the line: Draw a straight line through the points (0, 1) and (1, 3). This is the graph of the equation y = 2x + 1.

Example 2: From the graph, find the equation of the line that passes through the points (1, 4) and (3, 8).

Solution:

We are given two points, (1, 4) and (3, 8), and we need to find the equation of the line that passes through them.

1. Find the slope (m):

The slope is calculated using the formula:$m=\frac{y_2-y_1}{x_2-x_1}$

Substituting the points $(x_1,y_1)=(1,4)$ and $(x_2,y_2)=(3,8):$$m=\frac{8-4}{3-1}=\frac{4}{2}=2$

2. Find the y-intercept (b):

We can use the point (1, 4) and the slope m = 2 in the equation y = mx + b to find b.
Substituting y = 4, x = 1, and m = 2 into the equation: $4=2(1)+b\Rightarrow 4=2+b\Rightarrow b=2$

3. Write the equation:

Now that we have m = 2 and b = 2, the equation of the line is:

y = 2x + 2 

Example 3: Solve for y when x = 5 in the equation y = -3x + 4.

Solution:

Substitute x = 5 into the equation y = –3x + 4:

y = –3(5) + 4 = –15 + 4 = –11 

So, when x = 5, y = –11.

5.0Practice Questions on Linear Graphs

Question 1: Find the slope and y-intercept of the equation y = 5x - 7.

Question 2: Graph the equation y = –x + 3.

Question 3: Find the equation of the line that passes through the points (2, 5) and (4, 9).

Question 4: The equation of a line is given by y = 3x + 4. What is the value of y when x = 2?

Question 5: Solve for x in the equation 2x – 5y = 10 when y = 2.