Created By : Vaibhavi Kumari

Reviewed By : Phani Ponnapalli

Last Updated : Oct 10, 2023

Here is the handy online calculator that solves the nonlinear equations within seconds and displays the values of variables in the output field. All you have to do is enter the equation in the input box and tap on the calculate button in the Non Linear Equations Calculator and get the solutions effortlessly.

### What is meant by Linear Equation?

A nonlinear equation is an equation in which the degree of a term is 2 and more than 2. Curves are formed when non-linear equation values are plotted in the graph. The standard form of a nonlinear equation is ax² + by² = c, where a, b, c are constants and a0 and x and y are variables. Some of the instances of systems of non-linear equations are 2x²+ 3y² = 7, x² + 12xy + y² = 0.

### How to Solve Systems of Non Linear Equations?

There are three methods of solving nonlinear equations and they are explained as such with examples:

#### 1. Substitution Method:

One of the finest methods to solve linear and nonlinear equations is substitution. The procedure of using the substitution method in linear equations is similar to nonlinear equations. First of all, consider solving one equation and then substitute the result in the second equation to find another variable, and so on. However, you will get different variations in the possible results.

Steps To Follow:

1. Initially, understand the equations and start solving the equation for one of the variables.
2. Next, substitute the resulting expression in step one into the given nonlinear equation.
3. Find the remaining variables of the given equations.

Example:

Solve the system of equations: x − y = −1 and 2y = x² + 1

Solution:
Given systems of equations are:
x − y = −1  ------(1)
2y = x² + 1 ------(2)
Take the equation (1) and solve the x variable, we get
x - y = -1
x = y - 1
Now, substitute the resulting expression into the second equation
y = x² + 1 ------(2)
y = (y-1)² + 1
y = (y²-2y+1) +1
y = y² - 2y + 2
0 = y² - 3y + 2
0 =(y−2)(y−1)
y - 2 = 0
y = 2
And
y - 1 = 0
y = 1
Substitute the y values in equation (1) and solve the x value.
If y = 2 then x = y-1 = 2-1 = 1
If y = 1 then x = y-1 = 1-1 = 0
The solutions are (1,2) and (0,1), which can be verified by substituting these (x,y) values into both of the original equations.

#### 2. Elimination Method:

The method of elimination is the second-best approach for solving the nonlinear equations of two variables. The following steps will benefit you to grasp the process clearly and make you solve the values of the variables with ease.

Steps to Follow:

1. Firstly, check if the given equations have like variables of the second degree.
2. If the system is having the like variables then proceed with the elimination method by adding or subtracting.
3. In general, elimination is a quite easier way when it involves only two equations in two variables (a two-by-two system), instead of a three-by-three system, as there are fewer steps.

Find the simple instance equations below and examine the possible types of solutions.

Example:

Solve the system of nonlinear equations by elimination:
x² + y² = 1
2x² + 3y²  = 7

Solution:
Given nonlinear equations are
x² + y² = 1 ………(1)
2x² + 3y²  = 7 …..(2)
Now, consider equation (1) and multiply with 2
2(x²) + 2(y²) = 2(1)
2x² + 2y² = 2 ……..(3)
Subtract (3) with (2)
2x² + 3y² = 7
-2x² -2y² = -2
⇒y² = 5
⇒ y = √5 , y = -√5
Substitute the y values in equations (1)
If y = √5 then
x² + y² = 1
x² + 5  = 1
x²  = 1-5
x² = -4
x = 2i, x = −2i
If y = -√5 then x = 2i, x = −2i
The solutions for nonlinear equations of x² + y² = 1, 2x² + 3y²  = 7 are (x=2i, y=√5), (x=−2i, y=√5), (x=2i, y=−√5), (x=−2i, y=−√5).

### FAQs on System of Equations

1. How do non-linear equations form?

Generally, a non-linear equation is given by ax²+by² = c, where a,b,c are constant values and x, y are variables.

2. How does the graph of the nonlinear equation look?

A Non-linear equation graph represents curves and displays the variation in slope at different points.

3. What are the methods used to solve nonlinear equations?

The three methods of solving nonlinear equations are substitution, elimination, and graphing.

4. Can I find the non-linear equation calculator online?

Yes, you can find the nonlinear equations calculator online from Linearequationscalculator.com. It is a reliable and trustworthy online education portal that offers various free & user-friendly online calculator tools for math concepts.