Stewart Precalc 6e Section 10.8: Systems of Nonlinear Equations

Find all solutions of the system. \( \begin{cases} x^4 + y^3 = 17 \\ 3 x^4 + 5 y^3 = 53 \end{cases} \)

Find all solutions of the system. \( \begin{cases} \dfrac{2}{x} - \dfrac{3}{y} = 1 \\ -\dfrac{4}{x} + \dfrac{7}{y} = 1 \end{cases} \)

Find all solutions of the system. \( \begin{cases} \dfrac{4}{x^2} + \dfrac{6}{y^4} = \dfrac{7}{2} \\ \dfrac{1}{x^2} - \dfrac{2}{y^4} = 0 \end{cases} \)

Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. \( \begin{cases} y = x^2 + 8 x \\ y = 2 x + 16 \end{cases} \)

Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. \( \begin{cases} y = x^2 - 4 x \\ 2 x - y = 2 \end{cases} \)

Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. \( \begin{cases} x^2 + y^2 = 25 \\ x + 3 y = 2 \end{cases} \)

Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. \( \begin{cases} x^2 + y^2 = 17 \\ x^2 - 2 x + y^2 = 13 \end{cases} \)

Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. \( \begin{cases} \dfrac{x^2}{9} + \dfrac{y^2}{18} = 1 \\ y = -x^2 + 6 x - 2 \end{cases} \)

Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. \( \begin{cases} x^2 - y^2 = 3 \\ y = x^2 - 2 x - 8 \end{cases} \)

Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. \( \begin{cases} x^4 + 16 y^4 = 32 \\ x^2 + 2 x + y = 0 \end{cases} \)

Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. \( \begin{cases} y = e^x + e^{-x} \\ y = 5 - x^2 \end{cases} \)

**Dimensions of a Rectangle.** A rectangle has an area of \(180\) cm\(^2\) and a perimeter of \(54\) cm. What are its dimensions?

**Legs of a Right Triangle.** A right triangle has an area of \(84\) ft\(^2\) and a hypotenuse \(25\) ft long. What are the lengths of its other two sides?

**Dimensions of a Rectangle.** The perimeter of a rectangle is \(70\), and its diagonal is \(25\). Find its length and width.

**Dimensions of a Rectangle.** A circular piece of sheet metal has a diameter of \(20\) in. The edges are to be cut off to form a rectangle of area \(160\) in\(^2\) (see the figure). What are the dimensions of the rectangle?

**Flight of a Rocket.** A hill is inclined so that its 'slope' is \(\dfrac{1}{2}\). A coordinate system has its origin at the base of the hill, with axes scaled in meters. A rocket is fired from the base of the hill along a trajectory \(y = -x^2 + 401 x\). At what point does the rocket strike the hillside, and how far is this point from the base of the hill (to the nearest centimeter)?

**Making a Stovepipe.** A rectangular piece of sheet metal with an area of \(1200\) in\(^2\) is to be bent into a cylindrical length of stovepipe having a volume of \(600\) in\(^3\). What are the dimensions of the sheet metal?

**Global Positioning System (GPS).** The Global Positioning System determines the location of an object from its distances to satellites in orbit around the earth. In the simplified, two-dimensional situation shown in the figure, determine the coordinates of \(P\) from the fact that \(P\) is \(26\) units from satellite \(A\) and \(20\) units from satellite \(B\).

**Intersection of a Parabola and a Line.** On graph paper or using a graphing calculator, draw the parabola \(y = x^2\). Then draw the graphs of \(y = x + k\) for various values of \(k\). For what value of \(k\) is there exactly one intersection point? Make a conjecture about the values of \(k\) for which the following system has two solutions, one solution, and no solution, and prove your conjecture. \( \begin{cases} y = x^2 \\ y = x + k \end{cases} \)

**Some Trickier Systems.** Solve the systems.

Find all solutions of the system. \(\begin{cases} x^2 + y^2 = 100 \\ 3 x - y = 10 \end{cases}\)