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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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