# 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.

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich