In Exercises 27-30, find the solution set for each system by...

In Exercises $27-30,$ find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding all points of intersection. Check all solutions in both equations. $$\left\{\begin{array}{r} x^{2}+y^{2}=9 \\ x-y=3 \end{array}\right.$$

In Exercises $27-30,$ find the solution set for each system by graphing both of the system's equations in the same rectangular
coordinate system and finding all points of intersection. Check all solutions in both equations.
$$\left\{\begin{array}{r}
x^{2}+y^{2}=9 \\
x-y=3
\end{array}\right.$$

Key Concepts

Linear Equation

A linear equation in two variables typically takes the form ax + by = c, and its graph is a straight line. Linear equations are fundamental in algebra, and their intersection with other graphs, such as circles, provides a basis for solving various systems of equations. The slope-intercept form and point-slope form are common ways to represent linear equations and analyze their behavior.

Circle

In the context of coordinate geometry, a circle is represented by an equation of the form x² + y² = r², where r is the radius. This equation describes all points in the plane that are at a fixed distance (the radius) from a central point (usually the origin). Understanding the properties and graph of a circle is key when solving systems that include circular equations.

System of Equations

A system of equations consists of two or more equations that have common variables, and the goal is to find all sets of values for these variables that satisfy every equation simultaneously. This concept is central to algebra and is commonly used to model situations where multiple constraints must be met at once.

Graphical Method for Solving Systems

The graphical method involves plotting each equation on a coordinate system and determining the points where the graphs intersect. These intersection points represent the solutions to the system, as they satisfy all equations simultaneously. The method provides a visual understanding of the relationship between the equations and the nature of their solutions.

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