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Graphing equations is a valuable technique used to visually understand mathematical relationships. When graphing, we plot equations on a coordinate plane, allowing us to see solutions and intersection points. In this exercise, we deal with a circle and a straight line.

• Circular equations are shaped like loops where every point is equidistant from the center.
• Linear equations form straight lines, displaying constant rates of change.

To graph these, we start by rewriting equations so they appear in familiar forms. This helps us easily identify features like slope and intercept for lines or center and radius for circles.
By visually inspecting graphs, you can determine where these shapes intersect. These points are solutions that satisfy both equations simultaneously.

Understanding circle equations is key to solving systems involving them. A circle equation typically looks like \[ (x-a)^2 + (y-b)^2 = r^2 \] where

• \((a, b)\) is the center of the circle,
• \( r \) represents the radius.

In our problem, the initial equation \( x^2 + y^2 - 8y = -15 \) must be converted into the standard form. This involves completing the square on the terms to express it as \( (x-0)^2 + (y-4)^2 = 1 \).
This adjustment shows that our circle has a center at \((0, 4)\) with a radius of 1. It's important to know how to rearrange and identify these characteristics to accurately graph the circle on a coordinate plane.

Intersection points are critical in understanding how different graphs relate to one another. These are the points where two equations meet, providing solutions that satisfy both equations in a system.
Finding these points involves solving the equations simultaneously. For our circle and line, we replace the \( y \) in the circle equation with the expression \( y = 2x + 6 \) from the line equation.
After substituting, solving the resulting quadratic equation helps find the \( x \) values. Substituting these \( x \) back into the line equation gives the corresponding \( y \) values.
Ultimately, the intersection points \((-2.732, 0.536)\) and \((0.732, 7.464)\) illustrate where the line crosses the circle. Recognizing and solving for these points is vital in comprehensively solving such systems of equations.