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Chapter 13: Problem 31

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 2 x^{2}+y^{2}=18 \\ x y=4 \end{array}\right.$$

Short Answer

Expert verified
The solutions of the system are \((x,y)=( 2\sqrt{2}, \sqrt{2}), ( -2\sqrt{2}, -\sqrt{2}), (1,4), (-1,-4)\)

Step by step solution

01

Solve the linear equation for one variable

Looking at the second equation \(x y=4\), it can be solved for \(x\) or \(y\). For example, solve for \(x\), we will have \(x=\frac{4}{y}\) .
02

Substitute the expression for x into the first equation

Replace \(x\) in the first equation \(2 x^{2}+y^{2}=18\) with the expression we got from step 1 i.e. \(\frac{4}{y}\). This yields \(2 (\frac{4}{y})^{2} +y^{2}=18 \).
03

Solve the resulting equation for y

Solving the equation \(2 (\frac{4}{y})^{2} +y^{2}=18\) for \(y\), we have \(2 (\frac{16}{y^2}) +y^{2}=18\), then multiply each term by \(y^2\) to simplify the equation to \(32 + y^4 = 18 y^2\), then simplify to \(y^4 -18 y^2 +32=0\), the roots of which are \(y^2=2,16\), giving us \(y=\pm \sqrt{2}, \pm \sqrt{16} = \pm 4\)
04

Substitute y-values back to get x-values

Substitute \(y\) back into the second equation \(x = \frac{4}{y}\) to find values of \(x\). For \(y= \sqrt{2}, -\sqrt{2}\), we get \(x = \frac{4}{\sqrt{2}} ), -(\frac{4}{\sqrt{2}}) ) which simplifies to \(x = 2\sqrt{2}, -2\sqrt{2}\). For \(y= 4, -4\), we get \(x = 1, -1\).

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Most popular questions from this chapter

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola. Then graph the conic section. $$7 x^{2}-7 y^{2}=28$$

The equation of a parabola is given. Determine: a. if the parabola is horizontal or vertical. b. the way the parabola opens. c. the vertex. $$x=-(y+1)^{2}+4$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm graphing an equation that contains neither an \(x^{2}\) -term nor a \(y^{2}\) -term, so the graph cannot be a conic section.

The equation of a parabola is given. Determine: a. if the parabola is horizontal or vertical. b. the way the parabola opens. c. the vertex. $$y=x^{2}-4 x-1$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I noticed that depending on the values for \(A\) and \(B\), assuming that they are not both zero, the graph of \(A x^{2}+B y^{2}=C\) can represent any of the conic sections other than a parabola.
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