Question

Graph each equation of the system. Then solve the system to find the points of intersection. $$ \left\\{\begin{array}{r} x^{2}+y^{2}=8 \\ x^{2}+y^{2}+4 y=0 \end{array}\right. $$

Short answer

The points of intersection are (2, -2) and (-2, -2).

Step-by-step solution

- Identify and rearrange the equations

The given system of equations is: 1) \( x^2 + y^2 = 8 \) 2) \( x^2 + y^2 + 4y = 0 \). Rewrite the second equation to isolate the terms involving \(y\): \( x^2 + y^2 + 4y = 0 \rightarrow x^2 + (y^2 + 4y) = 0 \).

- Complete the square for the second equation

Complete the square for the terms involving \(y\) in the second equation: \( y^2 + 4y \) can be written as \( (y+2)^2 - 4 \). Thus, the second equation becomes: \( x^2 + (y+2)^2 - 4 = 0 \rightarrow x^2 + (y+2)^2 = 4 \).

- Rewrite the equations in standard form

Now, rewrite both equations in standard form: 1) \( x^2 + y^2 = 8 \) 2) \( x^2 + (y+2)^2 = 4 \). These are equations of circles.

- Find the centers and radii of the circles

For the first circle, \( x^2 + y^2 = 8 \), the center is (0, 0) and the radius is \( \sqrt{8} = 2\sqrt{2} \). For the second circle, \( x^2 + (y+2)^2 = 4 \), the center is (0, -2) and the radius is \( \sqrt{4} = 2 \).

- Graph the circles

Graph both circles on the coordinate plane. The first circle with center (0,0) and radius \( 2\sqrt{2} \), and the second circle with center (0,-2) and radius 2. Identify the points where the circles intersect.

- Solve the system algebraically

Solve the system algebraically to find the exact points of intersection. Subtract the second equation from the first: \((x^2 + y^2) - (x^2 + (y+2)^2) = 8 - 4\) \( y^2 - (y+2)^2 = 4 \) \( y^2 - (y^2 + 4y + 4) = 4 \) \( -4y - 4 = 4 \) \( -4y = 8 \) \( y = -2 \). Substitute \( y = -2 \) back into the first equation: \( x^2 + (-2)^2 = 8 \) \( x^2 + 4 = 8 \) \( x^2 = 4 \) \( x = \pm 2 \). The points of intersection are \( (2, -2) \) and \( (-2, -2) \).

Key concepts

Graphing Equations

When graphing equations, you plot their points on a coordinate plane. Every equation has a set of solutions that form a line or curve. In our system, we have two equations representing circles. The first equation is:
\( x^2 + y^2 = 8 \)
This represents a circle centered at the origin (0, 0) with a radius \( \text{radius} = \text{sqrt}(8) = 2\text{sqrt}(2) \).
To plot this circle:

• Find the center point (0, 0).
• Use your radius (approximately 2.83) and draw a circle around the center.

The second equation is:
\( x^2 + (y+2)^2 = 4 \)
This circle is centered at (0, -2) with a radius of 2.
To graph this circle:

• Locate the center (0, -2).
• Draw the circle with a radius of 2.

By graphing these circles, you will visually see the points where they intersect. These intersection points are the solutions to the system of equations.

Completing the Square

Completing the square is a method used to solve quadratic equations. It can also help to rewrite equations for graphing shapes like circles.
Let's take the second equation in our system:
\( x^2 + y^2 + 4y = 0 \)
We rearrange it to isolate the y terms:
\( x^2 + (y^2 + 4y) = 0 \)
To complete the square for \( y^2 + 4y \), we add and subtract 4:
\[ y^2 + 4y = (y+2)^2 - 4 \]
This makes our equation:
\[ x^2 + (y+2)^2 - 4 = 0 \]
Simplifying, we get:
\[ x^2 + (y+2)^2 = 4 \]
Now it's clear that this represents a circle centered at (0, -2) with a radius of 2. Completing the square helps us recognize and graph geometric shapes more easily.

Circles in Coordinate Geometry

In coordinate geometry, a circle's equation can be written in standard form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, (h, k) is the center, and r is the radius.
For our first circle:
\[ x^2 + y^2 = 8 \]
The center is (0, 0), and the radius is \( \text{sqrt}(8) = 2\text{sqrt}(2) \).
For the second circle:
\[ x^2 + (y+2)^2 = 4 \]
The center is (0, -2), and the radius is 2.
Graphing both circles on the same coordinate plane helps us find their intersection points.
In circles, the tricky part isn't recognizing the shape but finding those points accurately using both graphing and algebraic methods.

Algebraic Solution of Systems

To find the points where two circles intersect, we use algebra. We have:
1) \( x^2 + y^2 = 8 \)
2) \( x^2 + (y+2)^2 = 4 \)
We subtract the second equation from the first:

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