Problem 55 - College Algebra

Question

What is a system of nonlinear equations? Provide an example with your description.

Step-by-Step Solution

Answer

A system of nonlinear equations is a set of two or more equations where at least one is a nonlinear equation. For example: \[x^2 + y^2 = 25 \] and \[y = x^2 \]. They form a system where solutions are the intersection points of the two curves represented by these equations.

Step 1: Defining a System of Nonlinear Equations

A system of nonlinear equations is a set of two or more equations, at least one of which is not a linear equation. In general, a linear equation in two variables has the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. A nonlinear equation is one that has a degree higher than one, so it can take forms like \(ax^2 + by^2 = c\) or \(ab = xy\). A system of these equations implies that we have multiple such equations together and we attempt to find a common solution that satisfies all equations.

Step 2: Example of a System of Nonlinear Equations

Let's consider an example: \[x^2 + y^2 = 25 \] \[y = x^2 \] Here, we have a system of two nonlinear equations. The first equation is a circle with center at the origin and radius 5. The second equation is a parabola that opens upward. The solutions to the system are the points where these two curves intersect.