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Chapter 13: Problem 25

Solve each system by the addition method. $$\left\\{\begin{array}{l} x^{2}+y^{2}=25 \\ (x-8)^{2}+y^{2}=41 \end{array}\right.$$

Short Answer

Expert verified
The solutions to the system of equations are \(x=3, y=4\) and \(x=3, y=-4\).

Step by step solution

01

Simplify the Equations

The given equations are: \(x^{2}+y^{2}=25\) and \((x-8)^{2}+y^{2}=41\). To simplify, rewrite the second equation with simplified terms. This results in: \(x^{2}-16x+64+y^{2}=41\).
02

Subtract the first equation from the simplified second equation

Doing so will produce a simple linear equation. Let's subtract equation \(x^{2}+y^{2}=25\) from the simplified equation \(x^{2}-16x+64+y^{2}=41\), this results in \(-16x+64=16\).
03

Solve for x

Further simplifying \(-16x+64=16\), subtract 64 from both sides, resulting in \(-16x=-48\). Then, divide both sides by -16, resulting in \(x=3\).
04

Solve for y

With \(x=3\), substitute x in the first equation \(x^{2}+y^{2}=25\). This results in: \(3^{2}+y^{2}=25\) or \(y^{2}=16\). Solving for y, we get two possible values for y, that is, \(y=4\) or \(y=-4\).
05

State the solution

Therefore, the solutions to the system of equations are \(x=3, y=4\) and \(x=3, y=-4\).

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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. $$(x-1)^{2}+(y+2)^{2}=16$$

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. $$\left\\{\begin{array}{l}x=(y+2)^{2}-1 \\\ (x-2)^{2}+(y+2)^{2}=1\end{array}\right.$$

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. $$\left\\{\begin{array}{l}x=(y-3)^{2}+2 \\ x+y=5\end{array}\right.$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=-2 y^{2}+4 y-3$$

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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