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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

write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
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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