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11.3 Systems of Nonlinear Equations and Inequalities: Two Variables

Intersections of a line with a circle or parabola by substitution or elimination, and shaded regions for nonlinear inequalities.

Algebra and Trigonometry 2e, 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables

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27 exercises

11.3 Exercise 7

Solve the given system of equations by substitution.

3x-y=-22x2-y=0\begin{aligned}3 x - y & = & -2 \\ 2 x^{2} - y & = & 0\end{aligned}
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11.3 Exercise 9

Solve the system of nonlinear equations.

x2+y2=10x-3y=-10\begin{aligned}x^{2} + y^{2} & = & 10 \\ x - 3 y & = & -10\end{aligned}
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11.3 Exercise 11

Find the solution set for the given system of nonlinear equations.

4x2+y2=13x2+y2=10\begin{aligned}4 x^{2} + y^{2} & = & 13 \\ x^{2} + y^{2} & = & 10\end{aligned}
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11.3 Exercise 15

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.

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11.3 Exercise 17

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?

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11.3 Exercise 19

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.

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11.3 Exercise 21

Solve the system of nonlinear equations using substitution.

        y=x−3 x 2 + y 2 =9 \begin{matrix}\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } y = x −3 \\ x^{2} + y^{2} = 9\end{matrix}

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11.3 Exercise 23

Solve the system of nonlinear equations using substitution.

        y=−x x 2 + y 2 =9 \begin{matrix}\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } y = - x \\ x^{2} + y^{2} = 9\end{matrix}

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11.3 Exercise 25

Solve the system of nonlinear equations using elimination.

4 x 2 −9 y 2 =36 4 x 2 +9 y 2 =36 \begin{matrix}4 x^{2} −9 y^{2} = 36 \\ 4 x^{2} + 9 y^{2} = 36\end{matrix}

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11.3 Exercise 27

Solve the system of nonlinear equations using elimination.

2 x 2 +4 y 2 =4 2 x 2 −4 y 2 =25x−10 \begin{matrix}2 x^{2} + 4 y^{2} = 4 \\ 2 x^{2} −4 y^{2} = 25 x −10\end{matrix}

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11.3 Exercise 29

Solve the system of nonlinear equations using elimination.

x 2 + y 2 + 1 16 =2500 y=2 x 2 \begin{matrix}x^{2} + y^{2} + \frac{1}{16} = 2500 \\ y = 2 x^{2}\end{matrix}

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11.3 Exercise 31

Use any method to solve the system of nonlinear equations.

− x 2 +y=2 −x+y=2 \begin{matrix}- x^{2} + y = 2 \\ - x + y = 2\end{matrix}

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11.3 Exercise 33

Use any method to solve the system of nonlinear equations.

x 2 + y 2 =1          y=− x 2 \begin{matrix}x^{2} + y^{2} = 1 \\ \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } y = - x^{2}\end{matrix}

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11.3 Exercise 35

Use any method to solve the system of nonlinear equations.

9 x 2 +25 y 2 =225 (x−6) 2 + y 2 =1 \begin{matrix}9 x^{2} + 25 y^{2} = 225 \\ \left(x −6\right)^{2} + y^{2} = 1\end{matrix}

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11.3 Exercise 37

Use any method to solve the system of nonlinear equations.

2 x 3 − x 2 =y     x 2 +y=0 \begin{matrix}2 x^{3} - x^{2} = y \\ \textrm{ }\textrm{ }\textrm{ } x^{2} + y = 0\end{matrix}

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11.3 Exercise 39

Use any method to solve the nonlinear system.

x 2 − y 2 =9          x=3 \begin{matrix}x^{2} - y^{2} = 9 \\ \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } x = 3\end{matrix}

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11.3 Exercise 41

Use any method to solve the nonlinear system.

x 2 − y 2 =9    x−y=0 \begin{matrix}x^{2} - y^{2} = 9 \\ \textrm{ }\textrm{ }\textrm{ } x - y = 0\end{matrix}

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11.3 Exercise 43

Use any method to solve the nonlinear system.

− x 2 +y=2         2y=−x \begin{matrix}- x^{2} + y = 2 \\ \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 2 y = - x\end{matrix}

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11.3 Exercise 45

Use any method to solve the nonlinear system.

x 2 + y 2 =1         y 2 = x 2 \begin{matrix}x^{2} + y^{2} = 1 \\ \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } y^{2} = x^{2}\end{matrix}

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11.3 Exercise 47

Use any method to solve the nonlinear system.

     3 x 2 − y 2 =12 (x−1) 2 + y 2 =1 \begin{matrix}\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 3 x^{2} - y^{2} = 12 \\ \left(x −1\right)^{2} + y^{2} = 1\end{matrix}

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11.3 Exercise 49

Use any method to solve the nonlinear system.

3 x 2 − y 2 =12    x 2 + y 2 =16 \begin{matrix}3 x^{2} - y^{2} = 12 \\ \textrm{ }\textrm{ } x^{2} + y^{2} = 16\end{matrix}

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11.3 Exercise 51

Use any method to solve the nonlinear system.

x 2 + y 2 −6y=7           x 2 +y=1 \begin{matrix}x^{2} + y^{2} −6 y = 7 \\ \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } x^{2} + y = 1\end{matrix}

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11.3 Exercise 63

Find the solutions to the nonlinear equations with two variables.

6 x 2 − 1 y 2 =8 1 x 2 − 6 y 2 = 1 8 \begin{matrix}\frac{6}{x^{2}} - \frac{1}{y^{2}} = 8 \\ \frac{1}{x^{2}} - \frac{6}{y^{2}} = \frac{1}{8}\end{matrix}

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11.3 Exercise 65

Find the solutions to the nonlinear equations with two variables.

x 2 −xy−2 y 2 −6=0 x 2 + y 2 =1 \begin{matrix}x^{2} - x y −2 y^{2} −6 = 0 \\ x^{2} + y^{2} = 1\end{matrix}

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11.3 Exercise 67

Solve the system of inequalities. Use a calculator to graph the system to confirm the answer.

xy<1 y> x \begin{matrix}x y < 1 \\ y > \sqrt{x}\end{matrix}

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11.3 Exercise 69

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?

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11.3 Exercise 71

Construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.

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

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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.