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Chapter 13: Problem 30

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x+y^{2}=4 \\ x^{2}+y^{2}=16 \end{array}\right.$$

Short Answer

Expert verified
Therefore, the solutions are \((x, y) = (4, 0), (0, 2), (0, -2)\)

Step by step solution

01

Make \(x\) the subject in the first equation

Rearrange the first equation \(x + y^2 = 4\) to make \(x\) the subject. This gives \(x = 4 - y^2\).
02

Substitute for \(x\) in the second equation

Substitute \(x = 4 - y^2\) into the second equation \(x^2 + y^2 = 16\), yielding \((4 - y^2)^2 + y^2 = 16\), which simplifies to \(16 - 8y^2 + 2y^4 = 16\). After cancelling the 16 from both sides, we get \(2y^4 - 8y^2 = 0\). Factor out \(2y^2\), resulting in \(2y^2(y^2 - 4) = 0\), leading to \(2y^2 = 0\) and \(y^2 - 4 = 0\). Thus \(y = 0, \pm 2\).
03

Substitute \(y\) values in to the rearranged first equation

Substitute \(y = 0\) into the first rearranged equation \(x = 4 - y^2\) to get \(x = 4\). Then substitute \(y = 2\) into the first rearranged equation to get \(x = 0\). And lastly, substitute \(y = -2\) into the first rearranged equation to also get \(x = 0\).

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Most popular questions from this chapter

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

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.

Multiply: \(\quad(3 x-2)\left(2 x^{2}-4 x+3\right)\)

The towers of the Golden Gate Bridge connecting San Francisco to Marin County are 1280 meters apart and rise 140 meters 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 \((640,140)\) 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 640 for \(x\) and 140 for \(y\) and determining the value of \(a\). b. Use the equation in part (a) to find the height of the cable 200 meters from a tower. Round to the nearest meter.

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. $$y=2(x-3)^{2}+1$$
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