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Chapter 13: Problem 22

Solve each system by the addition method. $$\left\\{\begin{array}{l} 3 x^{2}-2 y^{2}=-5 \\ 2 x^{2}-y^{2}=-2 \end{array}\right.$$

Short Answer

Expert verified
The system of equations does not have a solution.

Step by step solution

01

Organize equations

Organize the two equations in terms of \(x^{2}\) and \(y^{2}\). The first equation can be written as \(3x^2 - 2y^2 = -5\) and the second as \(2x^2 - y^2 = -2\). These are our original equations.
02

Prepare for addition

Now we multiply the second equation by two to align the coefficients of \(y^2\) in both equations. This yields \( 4x^2 - 2y^2 = -4\).
03

Add and solve

Add the two equations: \(3x^2 + 4x^2 = -5 -4\). This gives \(7x^2 = -9...\Rightarrow x^2 = -\frac{9}{7}\).\ As \(x^2\) should be a real number, this gives a contradiction leading to the conclusion that the system of equations does not have a solution.
04

Test solution

As the result we got contradicts a property of real numbers (that their square is nonnegative), the system of equations does not have any real solution. We can stop here.

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Most popular questions from this chapter

Graph: \(f(x)=2^{1-x}\). (Section \(12.1,\) Example 4 )

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.

The George Washington Bridge spans the Hudson River from New York to New Jersey. Its two towers are 3500 feet apart and rise 316 feet above the road. The cable between the towers has the shape of a parabola and the cable just touches the sides of the road midway between the towers. The parabola is positioned in a rectangular coordinate system with its vertex at the origin. The point \((1750,316)\) lies on the parabola, as shown. (IMAGE CANT COPY) a. Write an equation in the form \(y=a x^{2}\) for the parabolic cable. Do this by substituting 1750 for \(x\) and 316 for \(y\) and determining the value of \(a\) b. Use the equation in part (a) to find the height of the cable 1000 feet from a tower. Round to the nearest foot.

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. $$y=x^{2}+6 x+10$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{l}x=2 y^{2}+4 y+5 \\\ (x+1)^{2}+(y-2)^{2}=1\end{array}\right.$$
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