# 11.3 Systems of nonlinear equations and inequalities: two variables

 Page 1 / 9
In this section, you will:
• Solve a system of nonlinear equations using substitution.
• Solve a system of nonlinear equations using elimination.
• Graph a nonlinear inequality.
• Graph a system of nonlinear inequalities.

Halley’s Comet ( [link] ) orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.

In this section, we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.

## Solving a system of nonlinear equations using substitution

A system of nonlinear equations    is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form $\text{\hspace{0.17em}}Ax+By+C=0.\text{\hspace{0.17em}}$ Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.

## Intersection of a parabola and a line

There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.

## Possible types of solutions for points of intersection of a parabola and a line

[link] illustrates possible solution sets for a system of equations involving a parabola and a line.

• No solution. The line will never intersect the parabola.
• One solution. The line is tangent to the parabola and intersects the parabola at exactly one point.
• Two solutions. The line crosses on the inside of the parabola and intersects the parabola at two points.

Given a system of equations containing a line and a parabola, find the solution.

1. Solve the linear equation for one of the variables.
2. Substitute the expression obtained in step one into the parabola equation.
3. Solve for the remaining variable.
4. Check your solutions in both equations.

## Solving a system of nonlinear equations representing a parabola and a line

Solve the system of equations.

$\begin{array}{l}x-y=-1\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y={x}^{2}+1\hfill \end{array}$

Solve the first equation for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and then substitute the resulting expression into the second equation.

Expand the equation and set it equal to zero.

Solving for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ gives $\text{\hspace{0.17em}}y=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=1.\text{\hspace{0.17em}}$ Next, substitute each value for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ into the first equation to solve for $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ Always substitute the value into the linear equation to check for extraneous solutions.

The solutions are $\text{\hspace{0.17em}}\left(1,2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(0,1\right),\text{}$ which can be verified by substituting these $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ values into both of the original equations. See [link] .

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
factoring polynomial
find general solution of the Tanx=-1/root3,secx=2/root3
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
where can I get indices
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Need help with this question please
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has