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Difference Between Linear and Nonlinear Equations

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Last updated date: 01st Mar 2024
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Introduction

We have learned about equations in the earlier classes. An equation is a statement of the equality of two expressions. The two sides of the equality sign are referred to as the left-hand side (LHS) and the right-hand side (RHS) of the equation. 

 

For example, in the equation 3x + 4 = 8, where 3, 4, and 8 are the constants, and x is the variable. The LHS is given by the expression 3x + 4 and the RHS is given by the constant 8. The equation remains unchanged if we carry out the same operation on both sides of the equation.

 

To solve an equation, we carry out a series of identical Mathematical operations on two sides of the equation such that the unknown variable is on one side and its value is obtained on the other side.

 

Equation: An equation is a statement of equality of two algebraic expressions involving constants and variables. 

 

Based on the degree and variable in the equations, they are classified as linear and nonlinear equations.


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Linear Equation

An equation in which the maximum degree of a term is one is called a linear equation. Or we can say that a linear equation that has only one variable is called a linear equation in one variable. 

 

A linear equation value when plotted on the graph forms a straight line.

The general form of a linear equation is ax + b = c, where a, b, c are constants and a0 and x and y are variables.

 

For Example: x + 7 = 12, 5/2x - 9 = 1, x2 + 1 = 5 and x/3 + 5 = x/2 - 3 are equations in one variable x. 

 

Here the highest power of each equation is one. 

 

2x + 3y = 15, 7x - y/3 = 3 are equations in two variables x and y.

 

When the linear equation is plotted on the graph we get the below figure.


Nonlinear Equation 

An equation in which the maximum degree of a term is 2 or more than two is called a nonlinear equation.

 

For example \[3x^{2}\] + 2x + 1 = 0, 3x + 4y = 5, this is the example of nonlinear equations, because equation 1 has the  highest degree of 2 and the second equation has variables x and y.

 

The nonlinear equation values when plotted on the graph forms a curve.

 

The general form of a nonlinear equation is \[ax^{2} + by^{2} = c\], where a, b, c are constants and a0 and x and y are variables.

 

When plotted on the graph we get the below curve


Difference Between Linear and Nonlinear Equations

Understanding the difference between linear and nonlinear equations is foremost important. Here is the table which will clarify the difference between linear and nonlinear equations. So let us understand exactly what linear and nonlinear equations are.

 

Differentiate Between Linear and Nonlinear Equations

Linear Equations

Non-Linear Equations

A Linear equation can be defined as the equation having a maximum of only one degree.

A Nonlinear equation can be defined as the equation having the maximum degree 2 or more than 2.

A linear equation forms a straight line on the graph.

A nonlinear equation forms a curve on the graph.

The general form of linear equation is,

y = mx +c

Where x and y are the variables, m is the slope of the line and c is a constant value.

The general form of nonlinear equations is,

ax2 + by2 = c

Where x and y are the variables and a,b and c are the constant values

Examples:

  • 3x + 2 = 5

  • 4x + 5 = 1

Examples:

  • 2x2+ 3y2 = 7

  • a2 + 2ab + b2 = 0

Let us understand what are linear and nonlinear equations with the help of some examples.


Solved Examples

Example1:  Solve the Linear equation 9(x + 1) = 2(3x + 8)

Solution:

9(x + 1) = 2(3x + 8)

Expand each side

9x + 9 = 6x + 16

Subtract 6x from both the sides

9x + 9 - 6x = 6x + 16 - 6x

3x + 9 = 16

Subtract 9 from both the sides

3x + 9 - 9 = 16 - 9

3x = 7

Divide each by 3

3x/3 = 7/3

x= 7/3

 

Example 2 : Solve the nonlinear equation 

\[3x^{2}\] - 5x + 2 = 0 

Solution: 

\[3x^{2}\] - 5x + 2 = 0

Factorizing

\[3x^{2}\] - 3x - 2x + 2 = 0

3x(x - 1) - 2(x - 1) = 0

( 3x - 2)( x - 1) = 0

(3x - 2) = 0 or (x - 1) = 0

x = 2/3 or x = 1


Quiz Time 

Q. Solve the following linear equation and find the value of x

  1. 3(5x + 6) = 3x - 2

  2. (2x +9)/5 = 5

 

Q. Solve the nonlinear equations

  1. \[7x^{2} = 8 - 10x\]

  2. \[3x^{2} - 4 = 5x\]

 

Knowing basic math is important not only in each of the standards but it is also important that you keep your foundation strong even when you are learning for any higher class Math. Now finding these concepts online is not a problem at all as you can find them all listed on Vedantu. Here you will get to know what is the Difference Between Linear and Nonlinear Equations and how to distinguish between them! Learning the concepts related to the linear equation and non-linear equation will help you solve a lot of problems in Algebra as well.


What Forms A Linear Equation?

The simplest form of a linear equation can be explained in the form y = a + bx where both a and b represent constants in an equation while there will be two variables that will be present. This forms the backbone of the linear equation. This linear equation when it is plotted on a graph paper will yield you a straight line with the line passing through the origin. It will have a constant slope value throughout the straight line that is passing through the origin. 


Distinguish Between Linear and Non-Linear in A Single Look

When talking about linear and nonlinear equations it is to be understood that the linear equations will have no exponents while the non-linear equations that are present will contain exponents raised to higher powers than 1.

FAQs on Difference Between Linear and Nonlinear Equations

1. How do I know that an equation is a linear or nonlinear equation?

To determine whether the given equation is linear we have to determine that a given equation is in the format

y = mx + c 

where m is the slope

 x and y are the variables 

c is the y-intercept.

For example y = 2x + 1, here the equation has the highest degree as one So it is a linear equation.

A nonlinear equation will not match this equation.

You can also test an equation is linear or nonlinear by plotting it on the graph.

If an equation gives a straight line then that equation is a linear equation.

Example: y = 2x + 1 is the equation can be represented on the graph as 

Here it represents a straight line so it is a linear equation.

2. How to solve a linear equation?

To solve a linear equation we use the idea of a balance to find the value of x. We have to keep both the right-hand side and left-hand side balance. We can maintain this status by performing the same operation by on both sides, such as adding subtracting, multiplying, or dividing by the same numbers. 


Here are the following steps to solve a linear equation:

Step 1: Start by moving all of the terms that contain a variable to the left-hand side of the equation.

Step 2:Move the terms that do not contain variables to the right-hand side of the equation. 

Step 3: Look at the variable and determine if there are any other operations being performed on it. you will get the value.

Step 4: Check your answer for accuracy. To do this, put the value back into the original equation.

3. What are some of the features of nonlinear equations?

Some of the features of the non-linear equations can be provided as follows:

A simple example of a nonlinear equation can be provided in the form: ax2 + by2 = c

  1. A non-linear equation unlike a linear equation will not be a straight line instead it will have a curved nature.

  2. It has a variable slope range on the length of the curve and hence it provides different values at different areas of the curve.

  3. The power of nonlinear equations will be 2 or higher and it is seen that as this power increases the curvature of the non-linear equation graph also increases.

  4. It is seen that the input data and output result of a nonlinear system are not directly related.

4. What are the benefits of finding out the difference Between Linear and Nonlinear Equations?

There are various benefits when you get to find out the difference Between Linear and Nonlinear Equations and can be provided as follows:

  1. The linear relation between two objects can be clearly understood if you have understood the concept of linear equations.

  2. It is seen that as there is an increase in one variable there is also either an increase or decrease in the other variable that is present. Hence it allows you to estimate how much degree it increases or reduces.

5. Is there a particular way in which the linear and nonlinear equations can be solved?

There is a particular way in which you can solve the linear and nonlinear equations and can be provided in the following steps:

  1. Initially solve the linear or non-linear equation that is provided.

  2. Now substitute the answer that you have in one of the variables to find out the value of the other variable.


In such a way you will find out both the variables that are provided in the linear and nonlinear equations. You can also check out the Vedantu revision notes where everything regarding the linear and nonlinear equations is given in proper detail.

6. How does Vedantu help in finding out more information about the Difference Between Linear and Nonlinear Equations?

Vedantu has provided detailed information regarding the Difference Between Linear and Nonlinear Equations that helps students solve various problems related to linear and nonlinear equations. This in turn also becomes a very helpful tactic to use during the exams. With Vedantu experts answering the questions related to the difference Between Linear and Nonlinear Equations and finding out more about the same to help students score well will help them gain a better understanding and score better marks. You can also solve some questions that are provided along with the notes of the difference Between Linear and Nonlinear Equations that will help you understand how they are solved.

7. Is finding out more about the Difference Between Linear and Nonlinear Equations beneficial for kids right from the lower classes?

Yes, it is highly beneficial to find out the difference Between Linear and Nonlinear Equations and understand it right from the lower classes as it prepares the students how to tackle the problems that arise with regards to the difference Between Linear and Nonlinear Equations. Apart from this benefit the students also stay one step ahead when it comes to understanding the basic concepts. When the students tend to learn the concepts related to the difference Between Linear and Nonlinear Equations they also tend to get a lot of time to revise many detailed studies that are related to that topic.