Problem 72 - Algebra and Trigonometry

Question

will help you prepare for the material covered in the next section. Graph \(x-y=3\) and \((x-2)^{2}+(y+3)^{2}=4\) in the same rectangular coordinate system. What are the two intersection points? Show that each of these ordered pairs satisfies both equations.

Step-by-Step Solution

Answer

The two intersection points can be found graphically. To verify, substitute these points into both original equations to ensure that they satisfy both.

Step 1: Plot the first equation

The first equation \(x-y=3\) is a linear equation. This can be rewritten as \(y=x-3\), indicating that the y-intercept is -3 and the slope is 1. This line can be drawn on the coordinate system accordingly.

Step 2: Plot the second equation

The second equation \((x-2)^{2}+(y+3)^{2}=4\) is a circle. The center of the circle is at the point (2,-3) and the radius is 2 (since 2^2 = 4). Draw this circle on the same coordinate system.

Step 3: Identify the intersection points

The two graphs intersect at two points. These can be found visually by following where the line and the circle intersect.

Step 4: Verify the intersection points

The two intersection points should be substituted back into both original equations in order to verify that they satisfy both equations.