SAT Math 7: Systems of Nonlinear Equations

Nonlinear system

A system is nonlinear when changes in its output are not proportional to changes in its input. Mathematically, at least one governing equation cannot be expressed as a linear combination of the unknown variables or functions.

How can a nonlinear equation be recognized from its unknown variables?

The unknown appears with a power other than 1, is multiplied by another unknown, or occurs inside a non-polynomial function such as $\sin x$, $e^x$, or $\ln x$. For example, $x^2+y=0$ and $\sin x+y=0$ are nonlinear.

How does a nonlinear system of equations differ from a system in which every equation is linear?

A nonlinear system contains several simultaneous equations with at least one nonlinear equation. The system may also contain some linear equations; only one nonlinear relation is needed for the entire system to be nonlinear.

Polynomial equation

A polynomial equation sets a polynomial expression equal to a constant or zero, such as $x^2+x-1=0$. Polynomial equations are nonlinear when the polynomial has degree greater than one.

What does solving a system of polynomial equations mean?

It means finding the common zeros: values of all variables that make every polynomial in the system equal to zero. Finding all such solutions can require specialized algorithms, such as Gröbner-basis methods.