# 9.3 Systems of nonlinear equations and inequalities: two variables  (Page 5/9)

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## Verbal

Explain whether a system of two nonlinear equations can have exactly two solutions. What about exactly three? If not, explain why not. If so, give an example of such a system, in graph form, and explain why your choice gives two or three answers.

A nonlinear system could be representative of two circles that overlap and intersect in two locations, hence two solutions. A nonlinear system could be representative of a parabola and a circle, where the vertex of the parabola meets the circle and the branches also intersect the circle, hence three solutions.

When graphing an inequality, explain why we only need to test one point to determine whether an entire region is the solution?

When you graph a system of inequalities, will there always be a feasible region? If so, explain why. If not, give an example of a graph of inequalities that does not have a feasible region. Why does it not have a feasible region?

No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.

If you graph a revenue and cost function, explain how to determine in what regions there is profit.

If you perform your break-even analysis and there is more than one solution, explain how you would determine which x -values are profit and which are not.

Choose any number between each solution and plug into $\text{\hspace{0.17em}}C\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}R\left(x\right).\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}C\left(x\right) then there is profit.

## Algebraic

For the following exercises, solve the system of nonlinear equations using substitution.

$\left(0,-3\right),\left(3,0\right)$

$\left(-\frac{3\sqrt{2}}{2},\frac{3\sqrt{2}}{2}\right),\left(\frac{3\sqrt{2}}{2},-\frac{3\sqrt{2}}{2}\right)$

For the following exercises, solve the system of nonlinear equations using elimination.

$\begin{array}{l}\hfill \\ 4{x}^{2}-9{y}^{2}=36\hfill \\ 4{x}^{2}+9{y}^{2}=36\hfill \end{array}$

$\left(-3,0\right),\left(3,0\right)$

$\begin{array}{l}{x}^{2}+{y}^{2}=25\\ {x}^{2}-{y}^{2}=1\end{array}$

$\begin{array}{l}\hfill \\ 2{x}^{2}+4{y}^{2}=4\hfill \\ 2{x}^{2}-4{y}^{2}=25x-10\hfill \end{array}$

$\left(\frac{1}{4},-\frac{\sqrt{62}}{8}\right),\left(\frac{1}{4},\frac{\sqrt{62}}{8}\right)$

$\begin{array}{l}{y}^{2}-{x}^{2}=9\\ 3{x}^{2}+2{y}^{2}=8\end{array}$

$\begin{array}{l}{x}^{2}+{y}^{2}+\frac{1}{16}=2500\\ y=2{x}^{2}\end{array}$

$\left(-\frac{\sqrt{398}}{4},\frac{199}{4}\right),\left(\frac{\sqrt{398}}{4},\frac{199}{4}\right)$

For the following exercises, use any method to solve the system of nonlinear equations.

$\left(0,2\right),\left(1,3\right)$

$\left(-\sqrt{\frac{1}{2}\left(\sqrt{5}-1\right)},\frac{1}{2}\left(1-\sqrt{5}\right)\right),\left(\sqrt{\frac{1}{2}\left(\sqrt{5}-1\right)},\frac{1}{2}\left(1-\sqrt{5}\right)\right)$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}9{x}^{2}+25{y}^{2}=225\hfill \\ {\left(x-6\right)}^{2}+{y}^{2}=1\hfill \end{array}$

$\left(5,0\right)$

$\left(0,0\right)$

For the following exercises, use any method to solve the nonlinear system.

$\left(3,0\right)$

No Solutions Exist

$\begin{array}{l}\hfill \\ -{x}^{2}+y=2\hfill \\ -4x+y=-1\hfill \end{array}$

No Solutions Exist

$\begin{array}{l}{x}^{2}+{y}^{2}=25\\ {x}^{2}-{y}^{2}=36\end{array}$

$\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right),\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right),\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right),\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

$\left(2,0\right)$

$\left(-\sqrt{7},-3\right),\left(-\sqrt{7},3\right),\left(\sqrt{7},-3\right),\left(\sqrt{7},3\right)$

$\left(-\sqrt{\frac{1}{2}\left(\sqrt{73}-5\right)},\frac{1}{2}\left(7-\sqrt{73}\right)\right),\left(\sqrt{\frac{1}{2}\left(\sqrt{73}-5\right)},\frac{1}{2}\left(7-\sqrt{73}\right)\right)$

## Graphical

For the following exercises, graph the inequality.

${x}^{2}+y<9$

${x}^{2}+{y}^{2}<4$

For the following exercises, graph the system of inequalities. Label all points of intersection.

$\begin{array}{l}{x}^{2}+y<1\\ y>2x\end{array}$

$\begin{array}{l}{x}^{2}+y<-5\\ y>5x+10\end{array}$

$\begin{array}{l}{x}^{2}+{y}^{2}<25\\ 3{x}^{2}-{y}^{2}>12\end{array}$

$\begin{array}{l}{x}^{2}-{y}^{2}>-4\\ {x}^{2}+{y}^{2}<12\end{array}$

$\begin{array}{l}{x}^{2}+3{y}^{2}>16\\ 3{x}^{2}-{y}^{2}<1\end{array}$

## Extensions

For the following exercises, graph the inequality.

$\begin{array}{l}\hfill \\ y\ge {e}^{x}\hfill \\ y\le \mathrm{ln}\left(x\right)+5\hfill \end{array}$

$\begin{array}{l}y\le -\mathrm{log}\left(x\right)\\ y\le {e}^{x}\end{array}$

For the following exercises, find the solutions to the nonlinear equations with two variables.

$\begin{array}{l}\frac{4}{{x}^{2}}+\frac{1}{{y}^{2}}=24\\ \frac{5}{{x}^{2}}-\frac{2}{{y}^{2}}+4=0\end{array}$

$\begin{array}{c}\frac{6}{{x}^{2}}-\frac{1}{{y}^{2}}=8\\ \frac{1}{{x}^{2}}-\frac{6}{{y}^{2}}=\frac{1}{8}\end{array}$

$\left(-2\sqrt{\frac{70}{383}},-2\sqrt{\frac{35}{29}}\right),\left(-2\sqrt{\frac{70}{383}},2\sqrt{\frac{35}{29}}\right),\left(2\sqrt{\frac{70}{383}},-2\sqrt{\frac{35}{29}}\right),\left(2\sqrt{\frac{70}{383}},2\sqrt{\frac{35}{29}}\right)$

No Solution Exists

## Technology

For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.

$\begin{array}{l}xy<1\\ y>\sqrt{x}\end{array}$

$x=0,y>0\text{\hspace{0.17em}}$ and $0

$\begin{array}{l}{x}^{2}+y<3\\ y>2x\end{array}$

## Real-world applications

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.

Two numbers add up to 300. One number is twice the square of the other number. What are the numbers?

12, 288

The squares of two numbers add to 360. The second number is half the value of the first number squared. What are the numbers?

A laptop company has discovered their cost and revenue functions for each day: $\text{\hspace{0.17em}}C\left(x\right)=3{x}^{2}-10x+200\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}R\left(x\right)=-2{x}^{2}+100x+50.\text{\hspace{0.17em}}$ If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.

2–20 computers

A cell phone company has the following cost and revenue functions: $\text{\hspace{0.17em}}C\left(x\right)=8{x}^{2}-600x+21,500\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}R\left(x\right)=-3{x}^{2}+480x.\text{\hspace{0.17em}}$ What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris