Question

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

Short answer

The points of intersection are (0,2) and (0,-2).

Step-by-step solution

Graph the First Equation

The first equation is a circle: \[ x^2 + y^2 = 4 \]. This is a circle centered at the origin (0,0) with a radius of 2. Plot this circle on the coordinate plane.

Simplify the Second Equation

The second equation is: \[ x^2 + 2x + y^2 = 0 \]. Rewrite this in a more familiar form by completing the square. First, group the x terms together: \[ x^2 + 2x \]. To complete the square: \[ x^2 + 2x + 1 \], we must add and subtract 1: \[ x^2 + 2x + 1 - 1 + y^2 = 0 \] becomes \[ (x+1)^2 + y^2 = 1 \]. This represents a circle centered at (-1,0) with radius 1.

Graph the Second Equation

Graph the circle \[ (x+1)^2 + y^2 = 1 \] on the same coordinate plane. It will be a circle centered at (-1,0) with a radius of 1.

Find Points of Intersection

The points of intersection between these two graphs represent the solutions to the system. Observe the graphs: The two circles intersect at points (0,2) and (0,-2).

Key concepts

Graphing Equations

To solve systems of equations graphically, we plot the equations on a coordinate plane and see where they intersect. Each equation can represent different shapes like lines, parabolas, or circles. By drawing these graphical representations, we can visually find the points where the graphs meet.

• Start by understanding the shape of each equation.
• For circles, use the center and radius to draw. For lines, use the slope and intercept.
• Points of intersections are the solutions to the system of equations.

This method is intuitive and visual, helping to see the relationships between the equations immediately.

Circles

Circles are a type of geometric shape that you will commonly graph in math problems. The general equation for a circle is \(x^2 + y^2 = r^2\), where \(r\) is the radius, and the center is at \( (0,0) \). If the circle is not centered at the origin, the equation takes the form \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h,k) \) is the center.
To graph a circle:

• Identify the center of the circle.
• Determine the radius.
• Plot the center on the coordinate plane.
• Mark points equal distance \(r\) away from the center in all directions and connect them smoothly.

In the exercise, the first equation \(x^2 + y^2 = 4\) is a circle centered at the origin with a radius of 2. The second equation, after completing the square, \((x+1)^2 + y^2 = 1\) is a circle centered at \(-1\) and \(0\) with a radius of 1.

Completing the Square

Completing the square helps to convert a quadratic equation into a form that easily identifies the conic section it represents, like circles, parabolas, or ellipses. This process involves rearranging the terms and adding the necessary constants to create a perfect square trinomial.
For example, let's complete the square for \(x^2 + 2x\):

• Group the x terms: \(x^2 + 2x\)
• Add and subtract 1 inside the equation to form a perfect square trinomial:
• \((x+1)^2 + y^2 = 1\).

Now, the equation \(x^2 + 2x + y^2 = 0\) becomes \((x+1)^2 + y^2 = 1\). By completing the square, we can identify that it represents a circle centered at \(-1,0)\) with a radius of 1. This makes it easier to graph and understand its relationship with other equations.

Points of Intersection

When solving systems of equations, finding the points of intersection is crucial. These are the coordinates where the graphs of the equations meet. With circles, the points of intersection are where two circular graphs cross each other.
To find the points of intersection:

• Graph each equation accurately on the same coordinate plane.
• Look for overlapping points where the graphs touch or cross each other.
• Check the coordinates of these points. They form the solution to the system.

In our exercise, the circles \(x^2 + y^2 = 4\) and \((x+1)^2 + y^2 = 1\) intersect at the points \((0,2)\) and \((0,-2)\). These coordinates provide the solutions to the system of equations.