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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

can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
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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