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Chapter 5: Problem 55

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.

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Most popular questions from this chapter

The weekly demand and supply models for a particular brand of scientific calculator for a chain of stores are given by the demand model \(N=-53 p+1600,\) and the supply model \(N=75 p+320 .\) In these models, \(p\) is the price of the calculator and \(N\) is the number of calculators sold or supplied each week to the stores. a. How many calculators can be sold and supplied at \(\$ 12\) per calculator? b. Find the price at which supply and demand are equal. At this price, how many calculators of this type can be supplied and sold each week?

In Exercises \(19-30,\) solve each system by the addition method. \(3 x+2 y=14\) \(3 x-2 y=10\)

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(9 x-3 y=12\) \(y=3 x-4\)

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x+y=4\\\ &y=3 x \end{aligned} $$

Promoters of a rock concert must sell at least 25,000 dollars tickets priced at 35 dollars and 50 dollars per ticket. Furthermore, the promoters must take in at least 1,025,000 dollars in ticket sales. Find and graph a system of inequalities that describes all possibilities for selling the 35 dollars tickets and the 50 dollars tickets.
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