Ever wondered how those straight lines on a graph can tell a whole story? That’s what a linear graph does! It’s a simple, powerful way to show the relationship between two or more quantities—basically, if the graph makes a straight line, you’re looking at a linear graph. Knowing how to plot these is super handy, especially if you're prepping for exams like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP Exams, PERT, Accuplacer, or even the MCAT. In this guide, we’ll break down exactly how to plot linear graphs for equations in one and two variables, and we’ll throw in some solved examples to make sure you totally get it. Let’s make those lines work for you on test day!

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

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.

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 in the form y = mx + c. This will give 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 is different from an increasing function in a way that the graph does not go up from left to right. But, the value of y in the graph remains positive with every changing value of x.

Consider the graph shown above. 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 forx = 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 above. 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 forx = 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.

Difference Between Line Graph and Linear Graph

Let us understand the difference between a linear graph and line graph, graphically.

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.

Linear Graph Solved Examples

Que 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)

Que 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 2: 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.

Conclusion

Linear graphs are more than just straight lines—they’re a super useful way to visualize relationships between variables. Whether you're solving real-life problems or tackling tricky math questions on exams like the SAT, ACT, or GRE, understanding how to plot and interpret linear graphs is key. Keep practicing those steps, and you'll be ready to confidently handle any graph-based question that comes your way on test day!