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Section exercises

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.

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

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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.

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

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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 C ( x ) and R ( x ) . If C ( x ) < R ( x ) , then there is profit.

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Algebraic

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

    x + y = 4 x 2 + y 2 = 9

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          y = x −3 x 2 + y 2 = 9

( 0 , −3 ) , ( 3 , 0 )

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          y = x x 2 + y 2 = 9

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          y = x x 2 + y 2 = 9

( 3 2 2 , 3 2 2 ) , ( 3 2 2 , 3 2 2 )

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          x = 2 x 2 y 2 = 9

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

4 x 2 −9 y 2 = 36 4 x 2 + 9 y 2 = 36

( −3 , 0 ) , ( 3 , 0 )

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x 2 + y 2 = 25 x 2 y 2 = 1

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2 x 2 + 4 y 2 = 4 2 x 2 −4 y 2 = 25 x −10

( 1 4 , 62 8 ) , ( 1 4 , 62 8 )

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y 2 x 2 = 9 3 x 2 + 2 y 2 = 8

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x 2 + y 2 + 1 16 = 2500 y = 2 x 2

( 398 4 , 199 4 ) , ( 398 4 , 199 4 )

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

−2 x 2 + y = −5      6 x y = 9

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x 2 + y = 2   x + y = 2

( 0 , 2 ) , ( 1 , 3 )

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x 2 + y 2 = 1            y = 20 x 2 −1

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x 2 + y 2 = 1            y = x 2

( 1 2 ( 5 −1 ) , 1 2 ( 1 5 ) ) , ( 1 2 ( 5 −1 ) , 1 2 ( 1 5 ) )

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2 x 3 x 2 = y             y = 1 2 x

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9 x 2 + 25 y 2 = 225 ( x −6 ) 2 + y 2 = 1

( 5 , 0 )

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x 4 x 2 = y    x 2 + y = 0

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2 x 3 x 2 = y      x 2 + y = 0

( 0 , 0 )

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

x 2 + y 2 = 9           y = 3 x 2

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x 2 y 2 = 9            x = 3

( 3 , 0 )

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x 2 y 2 = 9            y = 3

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x 2 y 2 = 9      x y = 0

No Solutions Exist

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x 2 + y = 2 −4 x + y = −1

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x 2 + y = 2           2 y = x

No Solutions Exist

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x 2 + y 2 = 25 x 2 y 2 = 36

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x 2 + y 2 = 1          y 2 = x 2

( 2 2 , 2 2 ) , ( 2 2 , 2 2 ) , ( 2 2 , 2 2 ) , ( 2 2 , 2 2 )

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16 x 2 −9 y 2 + 144 = 0                  y 2 + x 2 = 16

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       3 x 2 y 2 = 12 ( x −1 ) 2 + y 2 = 1

( 2 , 0 )

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       3 x 2 y 2 = 12 ( x −1 ) 2 + y 2 = 4

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3 x 2 y 2 = 12     x 2 + y 2 = 16

( 7 , −3 ) , ( 7 , 3 ) , ( 7 , −3 ) , ( 7 , 3 )

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x 2 y 2 6 x 4 y 11 = 0                     x 2 + y 2 = 5

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x 2 + y 2 −6 y = 7            x 2 + y = 1

( 1 2 ( 73 −5 ) , 1 2 ( 7 73 ) ) , ( 1 2 ( 73 −5 ) , 1 2 ( 7 73 ) )

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x 2 + y 2 = 6          x y = 1

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Graphical

For the following exercises, graph the inequality.

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

x 2 + y < −5 y > 5 x + 10

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x 2 + y 2 < 25 3 x 2 y 2 > 12

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x 2 y 2 > −4 x 2 + y 2 < 12

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x 2 + 3 y 2 > 16 3 x 2 y 2 < 1

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Extensions

For the following exercises, graph the inequality.

y e x y ln ( x ) + 5

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y log ( x ) y e x

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

4 x 2 + 1 y 2 = 24 5 x 2 2 y 2 + 4 = 0

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6 x 2 1 y 2 = 8 1 x 2 6 y 2 = 1 8

( −2 70 383 , −2 35 29 ) , ( −2 70 383 , 2 35 29 ) , ( 2 70 383 , −2 35 29 ) , ( 2 70 383 , 2 35 29 )

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x 2 x y + y 2 −2 = 0                 x + 3 y = 4

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x 2 x y −2 y 2 −6 = 0                 x 2 + y 2 = 1

No Solution Exists

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x 2 + 4 x y −2 y 2 −6 = 0                             x = y + 2

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Technology

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

x y < 1 y > x

x = 0 , y > 0 and 0 < x < 1 , x < y < 1 x

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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

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

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A laptop company has discovered their cost and revenue functions for each day: C ( x ) = 3 x 2 −10 x + 200 and R ( x ) = −2 x 2 + 100 x + 50. 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

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A cell phone company has the following cost and revenue functions: C ( x ) = 8 x 2 −600 x + 21,500 and R ( x ) = −3 x 2 + 480 x . What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

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Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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