# 7.3 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.

An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
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Abhi
I rally confuse this number And equations too I need exactly help
salma
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salma
Commplementary angles
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Sherica
im all ears I need to learn
Sherica
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Tamia
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Uday
hi
salma
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×