A straight line drawn on a graph is called a linear graph. The graph of a linear equation is a way to show the relationship between two quantities that change at a constant rate. When the points of a relation are plotted on the xy-plane and connected, they form a straight line, indicating a linear relationship. Such a graph helps in understanding how one quantity changes with respect to another and makes it easier to interpret, predict, and solve problems in mathematics, physics, and real-life situations. The graph of a linear equation is a simple yet powerful visual tool.

In this math article, we shall read about linear graphs and the ways to plot the graph for linear equations in one and two variables. Also, we will solve examples for better understanding of the concept.

What is a Linear Graph?

Graphical representation of a straight line is termed as a linear graph. This graph can be used to plot linear equations in one or two variables. A linear graph of two variables helps in finding the relation between two entries.

Let us understand this using an example:

Saira works for a company and gets $15 per hour. Her average daily expense is $50. She is concerned about her savings and wants to know the minimum number of hours she should work to have some savings in hand.

Representing time by t and income by l, the linear equation becomes, l = 15t.

In this graph, we can see that the value on the vertical axis is greater than the expense of $50.

The horizontal axis shows the minimum number of hours Saira must work to save some money.

It is observed that if she works for 5 hours daily, her daily earning will be $75, and she can easily save $25 per day.

Linear Graph Equation

As we know that a linear graph is a graphical representation of a straight line, the equation for a linear graph will be similar to the equation of a line. For a linear equation in one variable, the linear graph equation looks like AX = B, where A cannot be equal to zero.

For a linear equation in two variables, the linear graph equation will be written as AX + BY = C for real values of A, B, and C, where A, and B cannot be equal to zero.

For Example: The linear graph for equation x – 3y = 9, is given as

Graphing Linear Equations

Plotting linear equations on a graph grid is an easy task. We can plot linear equation in one and two variables using the following methods:

Graphing Linear equations in one variable

The graph of linear equations in one variable is a straight line that is either parallel to the x-axis or y-axis depending on the variable. For example, an equation in one variable that is x, the linear equation graph is parallel to the y-axis, and that in y, is parallel to the x-axis.

Let the given equation be x = 2, the graph is drawn as:

And for the equation y = 5, the linear equation graph in one variable is represented as:

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Graphing Linear Equation in Two Variable

A linear equation in two variables is a straight line graph. Let us plot a linear equation in two variables on a graph.

Let the equation be: x – 3y = 9.

We need to first make a table before actualling plotting the points satisfying the equation.

Now, according to the table, the points to be plotted on the graph are (0,-3), (3,-2), (9,0), (12,1).

This is the graphical representation of the equation x – 3y = 9.

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Steps for Plotting Linear Graph

Now that we know, we can plot linear equations on a graph and can represent it as a straight line. Some of the simple steps that are followed for plotting the graph are:

Let us understand this with an example:

For an equation in two variables 2x – y = 4, the steps followed are as follows:

Step 1: Convert the equation into the form y = mx + c. This gives us y = 2x − 4.

Step 2: Now we have to replace the value of x and find the subsequent values of y in order to create the coordinates.

Step 3: Starting with putting x = 0 in the above equation, we get y = -4. Similarly, we put values of x = 2, we get y = 0.

Step 4: We can take more values of x and find the subsequent values of y to get a set of coordinates for plotting a line.

The coordinates corresponding to the given values are: (0,-4), (2,0), (3,2), and (4,4).

Step 5: Plot these points on the coordinate plane and join the points to get the desired line.

This is the graphical representation of line 2x – y = 4.

Positive Linear Graph 

We know that any graph for an increasing function goes up from left to right. A positive linear graph always rises from left to right, showing that the function increases as x increases. But, the value of y in the graph remains positive with every changing value of x. Consider the graph shown below. It is an example of a positive linear graph. Here we can see that values of y are always positive. Let us cross-check the result. We can see that for x = 0, y = 4, and for x = 4, y = 2. As the graph has values of y as positive so it is a positive linear graph.

Negative Linear Graph

We know that a decreasing function has a negative slope and the graph goes down as it moves from left to right. But a negative linear graph is different from decreasing function. It is a graph that has values of y as negative. Consider the graph shown below. It is an example of a negative linear graph. Here we can see that values of y are always negative. Let us cross check the result. We can see that for x = 0, y =-12, and for x = 6, y =-2. As the graph has values of y as negative so it is a negative linear graph.

How to Draw Linear Graph

To draw a linear graph on graph paper, follow these steps:

1. Make a table of values. You can either choose random values for x and calculate the corresponding y, or put x = 0 to find y and y = 0 to find x.
2. Plot these (x, y) points on the graph paper. Make sure the scales of the x-axis and y-axis match the values in your table.
3. Draw a straight line passing through all the points.

How to Read Linear Graph

Understanding a linear graph involves looking at its key parts:

• Gradient (Slope): The slope shows how steep the line is. A positive gradient rises from left to right, while a negative gradient falls.
• Y-Intercept: This is where the line crosses the y-axis, showing the value of y when x = 0.
• Other Values: The graph can also be used to find other values of x or y easily.

Real-Life Applications of Linear Graphs

Linear graphs are widely used in real-life situations because they clearly show the relationship between two variables that change at a constant rate. Some important applications include:

1. Economics and Business: Linear graphs help visualise steady profit, loss, or revenue over time, making it easier to track performance and make predictions.
2. Physics: They are used to represent uniform acceleration or constant velocity of moving objects, helping in understanding motion and calculating related values.
3. Engineering and Design: Linear graphs assist in calibrating instruments or systems with known input-output relationships, ensuring accuracy in measurements and operations.
4. Everyday Data Analysis: They are useful for comparing trends, growth, or changes in simple datasets where the relationship between variables is constant.
5. Education: Linear graphs help students understand the concept of slope, intercepts, and linear relationships in mathematics and science.

Common Mistakes to Avoid When Plotting Linear Graphs

Plotting linear graphs may seem simple, but small errors can lead to incorrect results. Students often make mistakes in scaling, placing points, or drawing lines. Understanding these common errors helps ensure accurate and clear graphs.

• Incorrect Scaling: Using uneven or inconsistent scales on the axes can distort the graph. Always choose a clear, consistent scale.
• Plotting Points Wrongly: Misplacing points due to calculation or reading errors will give an incorrect line.
• Ignoring the Intercepts: Forgetting to mark the x-intercept or y-intercept can misrepresent the line’s position.
• Connecting Points Carelessly: Drawing a line that doesn’t pass through all plotted points can lead to errors.
• Not Labeling Axes: Failing to label the x-axis and y-axis can confuse anyone reading the graph.
• Overcrowding the Graph: Plotting too many points too close together makes the graph unclear.

Difference Between Line Graph and Linear Graph

A line graph and a linear graph are two different kinds of graphs. A line graph shows how data changes over time or across different categories. The line can bend or go up and down. A linear graph shows a straight line from a linear equation. It shows a steady, constant relationship between two variables.

In the above graphs, we can see that graph B is different from graph A.

Graph A is a linear graph and graph B is a line graph. The major difference between a line graph and a linear graph is that all the points in a linear graph lie on the same line, i.e. they are collinear. However, in a line graph, these points are not necessarily in a straight line.

Difference Between Linear and Non-Linear Graphs

Linear and non-linear graphs are two main types of graphs used to show how variables relate to each other. A linear graph always forms a straight line, showing a constant rate of change, while a non-linear graph can take various shapes like curves, circles, or parabolas, where the rate of change is not constant. Understanding the differences helps in analysing data accurately and choosing the right graph type.

Important Notes on Plotting Linear Graphs

To draw a linear graph, two points (x, y) are usually enough. However, using only two points may hide errors in calculating these values, as any two points can always form a straight line. It is recommended to plot an extra point to verify that the solutions of the given linear equation are correct.

The equation y = kx, where k is a real number, gives a horizontal line parallel to the X-axis.

The equation x = ky, where k is a real number, gives a vertical line parallel to the Y-axis.

Linear Graph Examples

Question 1: For a given equation 2x + y = 8, complete the following table:

Solution:

The points in the table have to satisfy the given equation 2x + y = 8, let us solve this one by one.

Case 1: Given that y = 8, we have to find x

Let us put the value of y in 2x + y = 8

2x + 8 = 8

2x = 0

x = 0

So the first point is (0,8)

Case 2: Given that x = -2, we have to find y

Let us put the value of x in 2x + y = 8 

2(-2) + y = 8

-4 + y = 8

y = 8 + 4 = 12

So the second point is (-2,12)

Case 3: Given that y = 0, we have to find x

Let us put the value of y in 2x + y = 8

2x + 0 = 8

2x = 8

x = 4

So the third point is (4,0)

Question 2: Rahul can drive a car at a speed of 30km/hr. Draw a distance-time graph for this. From the graph, find the time taken by Rahul to cover 75 km, and distance covered by him in 3.5 hrs.

Solution: Given that Rahul is driving on a constant speed of 30km/hr, So, we can say that:

Distance covered by Rahul in 1 hr is 30 km

Distance covered by Rahul in 2 hrs is 60 km

Distance covered by Rahul in 3 hr is 90 km,and so on.

So, the table for plotting the points on graph is:

So, the points to be plotted become : (1,30), (2,60), (3,90), (4,120).

Plotting these points and joining them give us a straight line as shown in the graph below:

Now, from the graph we can find the other two parts of the question.

Time taken by Rahul to cover 75 km is 2.5 hrs, and distance covered by Rahul in 3.5 hrs is 105km.

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.