Problem 69 - College Algebra Essentials

Question

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{aligned} (x-2)^{2}+(y+3)^{2} &=4 \\ y &=x-3 \end{aligned} $$

Step-by-Step Solution

Answer

The solution involves graphing both the circle and the linear equation on the same coordinate system, calculating their points of intersection, and verifying these points by substituting them into the original equations.

Step 1: Graphing the circle

The equation \((x-2)^{2}+(y+3)^{2}=4\) represents a circle with center at (2, -3) and a radius of 2. In order to graph this equation, mark the center on the graph and draw a circle around it with the appropriate radius.

Step 2: Graphing the line

The equation \(y=x-3\) represents a straight line which passes through the point (0, -3) and has a gradient of 1. To graph this, draw a line starting at the point (0, -3) and with a slope of 1.

Step 3: Finding the Points of Intersection

Intersecting points are obtained when the x and y coordinates satisfy both equations simultaneously. Solve the system of equations \((x-2)^{2}+(y+3)^{2}=4\) and \(y=x-3\), substituting \(x-3\) for \(y\) in the first equation. Solve the resulting equation for \(x\). Then, substitute the \(x\) values obtained into the second equation to find the respective \(y\) values.

Step 4: Verify the Points of Intersection satisfy both equations

To confirm that these ordered pairs are indeed solutions, substitute each of them into both original equations. If both equations hold true, this would mean the points of intersection are accurate.

Key Concepts

Graphing Equations

Graphing equations helps you visually see where different equations meet on a graph. It can provide a clear understanding of how different mathematical relationships behave. For this exercise, we are dealing with two equations: one of a circle and another of a line.

• The circle's equation: \((x-2)^{2}+(y+3)^{2}=4\)
• The line's equation: \(y=x-3\)

When you're graphing these, it's essential to understand what each component means, like the center and radius of the circle, or the slope and intercept of the line. Graphing these accurately opens the door to finding intersection points and analyzing relationships between them.

Intersection Points

Intersection points are where two graphs meet or cross each other on a coordinate plane. In mathematical terms, these are called solutions to the system of equations.
To find these, you'll need to solve both equations simultaneously. This involves:

• Substituting equations: Replace \(y\) in the circle's equation with the expression \(x-3\) from the line equation.
• Solving for \(x\): Find the values of \(x\) that satisfy the new equation.
• Finding corresponding \(y\) values: Substitute the found \(x\) values back into the line equation \(y = x - 3\).

Once you have \((x, y)\) pairs, ensure they satisfy both the original circle and line equations. This confirms they are indeed intersection points.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, allows us to plot equations and visually analyze their properties. It consists of two perpendicular axes: the x-axis (horizontal) and y-axis (vertical). Each point on this plane is represented as \((x, y)\).

• The center of the circle in this system is the point where both axes' values are identified, \((2, -3)\) in this case.
• Plotting the line involves a starting point and a slope, here starting at \((0, -3)\) with a slope of 1.

Learning how to use this coordinate system is crucial for tackling problems involving graphing and finding intersection points, as it provides the framework to visually interpret results.

Circle and Line Intersection

Understanding how a circle and a line can intersect is a key concept in geometry. There are typically three possibilities:

• No intersection: They do not meet.
• One point: The line is tangent to the circle, touching it at a single point.
• Two points: The line crosses through the circle.

For our exercise, we need to determine how they intersect by solving the system of equations involving a circle and a line:
By substituting \(y = x - 3\) into the circle's equation \((x-2)^{2}+(y+3)^{2}=4\), you derive a quadratic equation. Solving this gives you potential intersection points.
Each solution should then be verified by substitution back into the original equations to ensure correctness.