9.7E: Exercises
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Verbal
1) 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.
- Answer
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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.
2) When graphing an inequality, explain why we only need to test one point to determine whether an entire region is the solution?
3) 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?
- Answer
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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.
4) If you graph a revenue and cost function, explain how to determine in what regions there is profit.
5) 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.
- Answer
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Choose any number between each solution and plug into \(C(x)\) and \(R(x)\). If \(C(x)<R(x)\), then there is profit.
Algebraic
For the exercises 6-10, solve the system of nonlinear equations using substitution.
6) \(\begin{align*} x+y &= 4\\ x^2 + y^2 &= 9 \end{align*}\)
7) \(\begin{align*} y &= x-3\\ x^2 + y^2 &= 9 \end{align*}\)
- Answer
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\((0,-3)\), \((3,0)\)
8) \(\begin{align*} y &= x\\ x^2 + y^2 &= 9 \end{align*}\)
9) \(\begin{align*} y &= -x\\ x^2 + y^2 &= 9 \end{align*}\)
- Answer
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\(\left ( -\dfrac{3\sqrt{2}}{2},\dfrac{3\sqrt{2}}{2} \right )\), \(\left ( \dfrac{3\sqrt{2}}{2},-\dfrac{3\sqrt{2}}{2} \right )\)
10) \(\begin{align*} x &= 2\\ x^2 - y^2 &= 9 \end{align*}\)
For the exercises 11-15, solve the system of nonlinear equations using elimination.
11) \(\begin{align*} 4x^2 - 9y^2 &= 36\\ 4x^2 + 9y^2 &= 36 \end{align*}\)
- Answer
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\((-3,0)\), \((3,0)\)
12) \(\begin{align*} x^2 + y^2 &= 25\\ x^2 - y^2 &= 1 \end{align*}\)
13) \(\begin{align*} 2x^2 + 4y^2 &= 4\\ 2x^2 - 4y^2 &= 25x-10 \end{align*}\)
- Answer
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\(\left ( \dfrac{1}{4},-\dfrac{\sqrt{62}}{8} \right )\), \(\left ( \dfrac{1}{4},\dfrac{\sqrt{62}}{8} \right )\)
14) \(\begin{align*} y^2 - x^2 &= 9\\ 3x^2 + 2y^2 &= 8 \end{align*}\)
15) \(\begin{align*} x^2 + y^2+\dfrac{1}{16} &= 2500\\ y &= 2x^2 \end{align*}\)
- Answer
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\(\left ( -\dfrac{\sqrt{398}}{4},\dfrac{199}{4} \right )\), \(\left ( \dfrac{\sqrt{398}}{4},\dfrac{199}{4} \right )\)
For the exercises 16-23, use any method to solve the system of nonlinear equations.
16) \(\begin{align*} -2x^2+y &= -5\\ 6x-y &= 9 \end{align*}\)
17) \(\begin{align*} -x^2+y &= 2\\ -x+y &= 2 \end{align*}\)
- Answer
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\((0,2)\), \((1,3)\)
18) \(\begin{align*} x^2+y^2 &= 1\\ y &= 20x^2-1 \end{align*}\)
19) \(\begin{align*} x^2+y^2 &= 1\\ y &= -x^2 \end{align*}\)
- Answer
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\(\left ( -\sqrt{\dfrac{1}{2}(\sqrt{5}-1)},\dfrac{1}{2}\left (1-\sqrt{5} \right ) \right )\), \(\left ( \sqrt{\dfrac{1}{2}(\sqrt{5}-1)},\dfrac{1}{2}\left (1-\sqrt{5} \right ) \right )\)
20) \(\begin{align*} 2x^3-x^2 &= y\\ y &= \dfrac{1}{2} -x \end{align*}\)
21) \(\begin{align*} 9x^2+25y^2 &= 225\\ (x-6)^2+y^2 &= 1 \end{align*}\)
- Answer
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\((5,0)\)
22) \(\begin{align*} x^4-x^2 &= y\\ x^2+y &= 0 \end{align*}\)
23) \(\begin{align*} 2x^3-x^2 &= y\\ x^2+y &= 0 \end{align*}\)
- Answer
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\((0,0)\)
For the exercises 24-38, use any method to solve the nonlinear system.
24) \(\begin{align*} x^2+y^2 &= 9\\ y &= 3-x^2 \end{align*}\)
25) \(\begin{align*} x^2-y^2 &= 9\\ x &= 3 \end{align*}\)
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\((3,0)\)
26) \(\begin{align*} x^2-y^2 &= 9\\ y &= 3 \end{align*}\)
27) \(\begin{align*} x^2-y^2 &= 9\\ x-y &= 0 \end{align*}\)
- Answer
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No Solutions Exist
28) \(\begin{align*} -x^2+y &= 2\\ -4x+y &= -1 \end{align*}\)
29) \(\begin{align*} -x^2+y &= 2\\ 2y &= -x \end{align*}\)
- Answer
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No Solutions Exist
30) \(\begin{align*} x^2+y^2 &= 25\\ x^2-y^2 &= 36 \end{align*}\)
31) \(\begin{align*} x^2+y^2 &= 1\\ y^2 &= x^2 \end{align*}\)
- Answer
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\(\left ( -\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2} \right )\), \(\left ( -\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2} \right )\), \(\left ( \dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2} \right )\), \(\left ( \dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2} \right )\)
32) \(\begin{align*} 16x^2-9y^2+144 &= 0\\ y^2 + x^2 &= 16 \end{align*}\)
33) \(\begin{align*} 3x^2-y^2 &= 12\\ (x-1)^2 + y^2 &= 1 \end{align*}\)
- Answer
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\((2,0)\)
34) \(\begin{align*} 3x^2-y^2 &= 12\\ (x-1)^2 + y^2 &= 4 \end{align*}\)
35) \(\begin{align*} 3x^2-y^2 &= 12\\ x^2 + y^2 &= 16 \end{align*}\)
- Answer
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\((-\sqrt{7},-3)\), \((-\sqrt{7},3)\), \((\sqrt{7},-3)\), \((\sqrt{7},3)\)
36) \(\begin{align*} x^2-y^2-6x-4y-11 &= 0\\ -x^2 + y^2 &= 5 \end{align*}\)
37) \(\begin{align*} x^2+y^2-6y &= 7\\ x^2 + y &= 1 \end{align*}\)
- Answer
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\(\left ( -\sqrt{\dfrac{1}{2}(\sqrt{73}-5)},\dfrac{1}{2}\left (7-\sqrt{73} \right ) \right )\), \(\left ( \sqrt{\dfrac{1}{2}(\sqrt{73}-5)},\dfrac{1}{2}\left (7-\sqrt{73} \right ) \right )\)
38) \(\begin{align*} x^2+y^2 &= 6\\ xy &= 1 \end{align*}\)
Graphical
For the exercises 39-40, graph the inequality.
39) \(x^2+y<9\)
- Answer
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40) \(x^2+y^2<4\)
For the exercises 41-45, graph the system of inequalities. Label all points of intersection.
41) \(\begin{align*} x^2 + y &<1 \\ y &>2x \end{align*}\)
- Answer
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42) \(\begin{align*} x^2 + y &<-5 \\ y &>5x+10 \end{align*}\)
43) \(\begin{align*} x^2 + y^2 &<25 \\ 3x^2 - y^2 &>12 \end{align*}\)
- Answer
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44) \(\begin{align*} x^2 - y^2 &>-4 \\ x^2 + y^2 &<12 \end{align*}\)
45) \(\begin{align*} x^2 + 3y^2 &>16 \\ 3x^2 - y^2 &<1 \end{align*}\)
- Answer
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Extensions
For the exercises 46-47, graph the inequality.
46) \(\begin{align*} y &\geq e^x \\ y &\leq \ln (x)+5 \end{align*}\)
47) \(\begin{align*} y &\leq -\log (x)\\ y &\leq e^x \end{align*}\)
- Answer
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For the exercises 48-52, find the solutions to the nonlinear equations with two variables.
48) \(\begin{align*} \dfrac{4}{x^2} + \dfrac{1}{y^2} &= 24\\ \dfrac{5}{x^2} - \dfrac{2}{y^2} + 4 &= 0 \end{align*}\)
49) \(\begin{align*} \dfrac{6}{x^2} - \dfrac{1}{y^2} &= 8\\ \dfrac{1}{x^2} - \dfrac{6}{y^2} &= \dfrac{1}{8} \end{align*}\)
- Answer
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\(\left ( -2\sqrt{\dfrac{70}{383}},-2\sqrt{\dfrac{35}{29}} \right )\), \(\left ( -2\sqrt{\dfrac{70}{383}},2\sqrt{\dfrac{35}{29}} \right )\), \(\left ( 2\sqrt{\dfrac{70}{383}},-2\sqrt{\dfrac{35}{29}} \right )\), \(\left ( 2\sqrt{\dfrac{70}{383}},2\sqrt{\dfrac{35}{29}} \right )\)
50) \(\begin{align*} x^2 - xy + y^2 - 2 &= 0\\ x+3y &= 4 \end{align*}\)
51) \(\begin{align*} x^2 - xy - 2y^2 - 6 &= 0\\ x^2 + y^2 &= 1 \end{align*}\)
- Answer
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No Solution Exists
52) \(\begin{align*} x^2 + 4xy - 2y^2 - 6 &= 0\\ x &= y+2 \end{align*}\)
Technology
For the exercises 53-54, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.
53) \(\begin{align*} xy &< 1\\ y &> \sqrt{x} \end{align*}\)
- Answer
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\(x=0\), \(y>0\) and \(0<x<1\), \(\sqrt{x} < y < \dfrac{1}{x}\)
54) \(\begin{align*} x^2 + y &< 3\\ y &> 2x \end{align*}\)
Real-World Applications
For the exercises 55-, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.
55) Two numbers add up to \(300\). One number is twice the square of the other number. What are the numbers?
- Answer
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\(12,288\)
56) The squares of two numbers add to \(360\). The second number is half the value of the first number squared. What are the numbers?
57) A laptop company has discovered their cost and revenue functions for each day: \(C(x)=3x^2-10x+200\) and \(R(x)=-2x^2+100x+50\). 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.
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\(2\) - \(20\) computers
58) A cell phone company has the following cost and revenue functions: \(C(x)=8x^2-600x+21,500\) and \(R(x)=-3x^2+480x\). What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.