Question

What is the minimum and maximum number of solutions that we can expect to see in any given system of nonlinear equations? In your own words, what is the meaning of extraneous solutions? When solving a system of nonlinear equations, is it possible to always use the Addition Method? Explain your reasoning in complete sentences. PLEASE TYPE, DO NOT WRITE IT DOWN and Check your punctuation and proofreading.

Answer 1

Q1.What are the minimum and the maximum number of solutions that we can expect to see in any given system of nonlinear equations?

Answer: A system of equations is a non-linear system of equations if at least one equation is non-linear. A system of non-linear equations can have zero solutions if the graphs of the equations do not intersect.

So,minimum:0 solutions

Maximum: can have an infinite number of solutions

Let's have an example:

Look at the graph below. Point A and B where the system of these two equations can have an infinite number of solutions

Q2.what is the meaning of extraneous solutions?

Answer: Extraneous solution some times called spurious solution

It evolves during the solving process of a problem but it's not a valid solution.

Let's have an example:

x+2=0

Note: In algebra, we can multiply both sides of an equation by the same expression without changing the equation's solutions.

Multiplying both sides of the above equation by x we get

x(x+2)=x(0)

x^2 +2x=0,this quadratic equation has -2 and 0 as solutions. But if we substitute x=0 in the equation x+2=0 we will get 2=0 and it's an invalid equation.

Q3.When solving a system of nonlinear equations, is it possible to always use the Addition Method?

Answer: No, it is not.

Let's have an example:

x+y=0 .............(1)

x^2+y^2=4 ..........(2)

The above two equations can not be solved simultaneously using the Addition Method. If we add the above two equations, then it will not be possible to cancel out one variable because one equation is a quadratic while the other one is the linear one.