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Chapter 13: Problem 20

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

Short Answer

Expert verified
The solution to the system of equations is \((x, y) = {(1, 0), (-1, 0)}\).

Step by step solution

01

Addition of given equations

Add the two equations together.\(4x^{2} - y^{2} + 4x^{2} + y^{2} = 4 + 4\). The y variable gets cancelled out, and the equation simplifies to \(8x^{2} = 8\).
02

Solve for x

Divide both sides of the equation by 8 to isolate x. This results in \(x^{2} = 1\). Then, take the square root of both sides which yields \(x = \pm1\).
03

Solve for y

Substitute x values into one of the original equations. Using the first original equation \(- y^{2} = 4 - 4x^{2}\). Substitute \(x = 1\) into the equation to get \(-y^{2} = 0\). Thus, \(y = 0\). Substitute \(x = -1\) into the equation to get \(-y^{2} = 0\). Thus, \(y = 0\) again.

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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. $$3 x^{2}=12+3 y^{2}$$

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.

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