# Conics

## Solve Systems of Nonlinear Equations


### Solve a System of Nonlinear Equations Using Graphing

We learned how to solve systems of linear equations with two variables by graphing, substitution and elimination. We will be using these same methods as we look at nonlinear systems of equations with two equations and two variables. A system of nonlinear equations is a system where at least one of the equations is not linear.
For example each of the following systems is a system of nonlinear equations.
Just as with systems of linear equations, a solution of a nonlinear system is an ordered pair that makes both equations true. In a nonlinear system, there may be more than one solution. We will see this as we solve a system of nonlinear equations by graphing.
When we solved systems of linear equations, the solution of the system was the point of intersection of the two lines. With systems of nonlinear equations, the graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions. Once you identify the graphs, visualize the different ways the graphs could intersect and so how many solutions there might be.
To solve systems of nonlinear equations by graphing, we use basically the same steps as with systems of linear equations modified slightly for nonlinear equations. The steps are listed below for reference.
To identify the graph of each equation, keep in mind the characteristics of the  and  terms of each conic.

### Solve a System of Nonlinear Equations Using Substitution

The graphing method works well when the points of intersection are integers and so easy to read off the graph. But more often it is difficult to read the coordinates of the points of intersection. The substitution method is an algebraic method that will work well in many situations. It works especially well when it is easy to solve one of the equations for one of the variables.
The substitution method is very similar to the substitution method that we used for systems of linear equations. The steps are listed below for reference.
So far, each system of nonlinear equations has had at least one solution. The next example will show another option.

### Solve a System of Nonlinear Equations Using Elimination

When we studied systems of linear equations, we used the method of elimination to solve the system. We can also use elimination to solve systems of nonlinear equations. It works well when the equations have both variables squared. When using elimination, we try to make the coefficients of one variable to be opposites, so when we add the equations together, that variable is eliminated.
The elimination method is very similar to the elimination method that we used for systems of linear equations. The steps are listed for reference.
There are also four options when we consider a circle and a hyperbola.

### Use a System of Nonlinear Equations to Solve Applications

Systems of nonlinear equations can be used to model and solve many applications. We will look at an everyday geometric situation as our example.

### Key Concepts

1. How to solve a system of nonlinear equations by graphing.


2. How to solve a system of nonlinear equations by substitution.


3. How to solve a system of equations by elimination.



### Section Exercises


### Practice Makes Perfect

Solve a System of Nonlinear Equations Using Graphing
In the following exercises, solve the system of equations by using graphing.
Solve a System of Nonlinear Equations Using Substitution
In the following exercises, solve the system of equations by using substitution.
Solve a System of Nonlinear Equations Using Elimination
In the following exercises, solve the system of equations by using elimination.
Use a System of Nonlinear Equations to Solve Applications
In the following exercises, solve the problem using a system of equations.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

### Chapter Review Exercises


### Distance and Midpoint Formulas; Circles

Use the Distance Formula
In the following exercises, find the distance between the points. Round to the nearest tenth if needed.
Use the Midpoint Formula
In the following exercises, find the midpoint of the line segments whose endpoints are given.
Write the Equation of a Circle in Standard Form
In the following exercises, write the standard form of the equation of the circle with the given information.
Graph a Circle
In the following exercises, ⓐ find the center and radius, then ⓑ graph each circle.

### Parabolas

Graph Vertical Parabolas
In the following exercises, graph each equation by using its properties.
In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.
Graph Horizontal Parabolas
In the following exercises, graph each equation by using its properties.
In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.
Solve Applications with Parabolas
In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.

### Ellipses

Graph an Ellipse with Center at the Origin
In the following exercises, graph each ellipse.
Find the Equation of an Ellipse with Center at the Origin
In the following exercises, find the equation of the ellipse shown in the graph.
Graph an Ellipse with Center Not at the Origin
In the following exercises, graph each ellipse.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
Solve Applications with Ellipses
In the following exercises, write the equation of the ellipse described.

### Hyperbolas

Graph a Hyperbola with Center at 
In the following exercises, graph.
Graph a Hyperbola with Center at 
In the following exercises, graph.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola
In the following exercises, identify the type of graph.

### Solve Systems of Nonlinear Equations

Solve a System of Nonlinear Equations Using Graphing
In the following exercises, solve the system of equations by using graphing.
Solve a System of Nonlinear Equations Using Substitution
In the following exercises, solve the system of equations by using substitution.
Solve a System of Nonlinear Equations Using Elimination
In the following exercises, solve the system of equations by using elimination.
Use a System of Nonlinear Equations to Solve Applications
In the following exercises, solve the problem using a system of equations.

### Practice Test

In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.
In the following exercises, write the standard form of the equation of the circle with the given information.
In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and ⓑ graph the equation.
In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, ⓑ write the equation in standard form, and ⓒ graph the equation.