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How to solve a non-linear equation

Before proceeding to non linear equations let’s understand what is a “system of equations”?  A system of equations is a set of different equations that can be interrelated to reach a final solution. Now this set or system of equations can be of linear equations too.  So what is a non-linear system of equations?
A non-linear system of equation involves two variables and is a set or system of equations in which at least one variable is raised to an exponent of value more than 1 and/or the two variables are multiplied in any one of the equation i.e. there is a product of the two variables involved in any of the equation from the given system. Ex:
(x² ~ y² = 20) & (2x – y = 10)

So, how to solve this non-linear system of equations?

There are two possible methods of solving the non-linear system of equations. One is the substitution method and the other is the elimination method.  We will see both the methods in details one by one.

Substitution method:

As you can realize from the name, here we will substitute one variable with the other to convert the set of non-linear equations into a single mono-variable (of only one variable) equation. What and how will we substitute? Well an example will make it clear but for the theory of it, we will derive an equation representing the value of one variable in terms of the other variable from any one of the given equations. That value when substituted in place of the variable in the other equation will give us a single equation in terms of a single variable. Ex:

Equation 1 à X² + Y² = 20; Equation 2 à 2X + Y = 2
From eq.2 we get, Y=2–2X
This when substituted in eq.1 gives us:   X² + (2~2X)² = 20;
i.e.   X² + 4 ~ 4X + 4X² = 20
5X2 ~4X ~ 16 = 0, which now can be solved as a quadratic equation.


Elimination method:

This method, again as implied by the name is targeted to eliminate one variable after applying a mathematical operation such as addition or subtraction to the set or system of non-linear equations. This will eventually result in a single equation expressed in terms of a single variable that can be easily solved. We will see an example to clarify it better.

Equation 1 à 4X² + 2Y² = 48; Equation 2 à X² ~ Y² = 12
Eq.2 multiplied by 2 on both sides; 2X² ~ 2Y² = 24
Now, Eq.2 is added to Eq.1;        [4X² + 2Y² = 48] + [2X² ~ 2Y² = 24] implies 6X² = 24
i.e.         X² = 24/6 = 4 i.e. X = √4 = ±2

So the above examples explain how you can solve a set of non-linear equations. We have to keep in mind that elimination method has some restriction and can’t be used on every set of non-linear equations as the power of the variables need to be same in both the equations and also should be separate entities and not product of two variables in both the equations.