Linear Graphs

The linear graph is a fundamental concept of mathematics that helps visualise the relation between two variables clearly and easily. Whether you're monitoring your spending over time, quantifying speed in physics, or modelling trends in data, linear graphs give you a neat yet effective means of representation. Linear graphs are based on the equations representing a straight-line relation. Here, we will be understanding these equations and how they are represented on the graph. 

1.0Linear Graph Definition

A linear graph is a graph that shows a straight-line relationship between two variables. In mathematics, it is a scenario in which the rate of change between the variables is constant. It implies that whenever one variable goes up or down, the other goes up or down at the same rate, creating a straight line when plotted on a coordinate plane. The straight line can slope upward, downward, or horizontal, based on the relationship between the variables.

2.0Linear Graph Equations 

Equations of a linear graph are the algebraic expressions of the relationship between two variables, generally x and y. Linear graph equations generally possess two core components, which are; the slope or gradient and the y-intercept.  

The linear equations can be represented in different forms as per the requirements of the question or the given information. These include: 

Slope-Intercept Form 

This form of linear equation is also known as the general form of a linear graph. It can be expressed as: 

y=m x+c

Here, 

• m is the slope of the equation 
• c is the y-intercept 

Standard Equation for Linear Graph

Every linear equation, even before it is plotted on the graph, is written in its standard form, which is: 

A x+B y=C

Here, 

• A and B are the coefficients of variables x and y. 
• C is the constant term in the equation. 

Point-slope form

This equation is used when any one coordinate and the slope of the line are given. This equation can be expressed as: 

$y-y_1=m\left(x-x_1\right)$

Here, 

• x₁ and y₁ are the known coordinates of a point on the line. 
• m is the slope. If two points (x₁, y₁) and (x₂, y₂) are known, then the slope can be calculated as: 

$m=\frac{y_2-y_1}{x_2-x_1}$

Or the point-slope form can be rewritten as: 

$y-y_1=\left(\frac{y_2-y_1}{x_2-x_1}\right)\left(x-x_1\right)$

3.0Drawing a Linear Graph 

Follow these simple steps to draw a linear graph on the graph paper: 

1. Create a table of values, choosing random values of x or y and calculate the corresponding value of y or x using the equation. Or 

You can also put x = 0 and write the value of y, and then put y = 0 and write the corresponding value of x to make the table. 

2. Plot these values of (x, y) on the graph paper. Remember to choose the values of the x and y axes according to the values calculated above. 
3. Now, draw a straight line passing through the points. 

4.0Reading a Linear Graph 

Interpreting or reading a linear graph basically involves understanding the components of a graph, which are: 

1. The Gradient (or Slope): The slope helps us understand how steep the line is. The steeper the line, the higher the gradient. If the gradient is positive, the line increases from left to right. If it is negative, the line slopes down. 
2. The Y-Intercept: This is where the line intersects the y-axis. It indicates the value of y at x = 0. This helps us know where the line begins on the graph.
3. Find other values of x or y: The linear graph also helps in finding other values of x and y. 

5.0Linear and Non-Linear Graphs 

Linear Graphs	Non-Linear Graphs
Linear graphs always form a straight line when represented on the graph.	These graphs are not straight lines; they can be any shape, like a parabola, a circle, or an ellipse.
The relation between variables always follows a constant rate of change.	The relation between variables is not necessarily proportional.
The equation for the graph always follows a standard form.	The equations for non-linear graphs may include powers, roots, or even trigonometric functions.

6.0Applications of Linear Graphs

A linear graph is used in numerous real-world situations due to its ability to easily visualise the relation between two variables changing at a constant rate. These applications are: 

1. The most effective use of linear Graphs is in economics and business to visualise a steady profit or loss over time. 
2. In physics, linear graphs are used to represent the uniform acceleration or constant velocity of a moving object. 
3. In engineering and design, these graphs are used to calibrate instruments with known input-output relationships. 

7.0Solved Examples of Linear Graphs

Problem 1: Draw a graph of the linear equation: y = x + 3. 

Solution: To draw the graph, first, create a table of values: 

Put y = 0, 

0 = x + 3 

x = –3

Now, put x = 0 

y = 0 + 3 

y = 3

Problem 2: Find the equation of the line that passes through the points (1, 4) and (3, 8). Also, draw the graph for the linear equation formed.  

Solution: For forming the equation, first find the slope of the equation “m” 

$\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{8-4}{3-1}=\frac{4}{2}=2\end{array}$

Now, form the equation: 

$\begin{array}{c}y-4=2(x-1)\\ y-4=2x-2\\ y=2x+2\end{array}$

For the graph, put x = 0 

y = 2(0) + 2 

y = 2 

Put y = 0 

0 = 2x + 2 

x = –1