What is the main purpose of solving a system of equations?

The primary purpose of solving a system of equations is to determine the points of intersection between the equations, which represents values that satisfy all equations in the system simultaneously. When two or more equations are graphed in the same coordinate plane, their intersection points indicate where they have common solutions. These points can be useful in various applications, such as finding equilibrium points in economics, determining optimal solutions in linear programming, or analyzing relationships in physics.

Other options focus on different aspects of mathematical analysis. Finding the maximum value of a function pertains to optimization, which is a separate task. Graphing the equations can help visualize the solutions but does not directly provide the solutions themselves, while finding the slope of each equation relates to understanding their rates of change rather than identifying common solutions. Thus, the correct answer accurately reflects the fundamental goal of solving a system of equations.