A system of equations is a collection of two or more equations that are solved simultaneously to find the values of the variables that satisfy all the equations.
The purpose of solving a system of equations is to find the unique solution(s) that satisfy all the equations in the system. This can help in solving real-world problems, such as finding the intersection point of two lines or determining the amount of ingredients needed for a recipe.
There are several methods for solving a system of equations, including substitution, elimination, and graphing. These methods involve manipulating the equations in the system to eliminate one of the variables and then solving for the remaining variable.
A system of equations has a solution if the equations are consistent, meaning they have at least one common solution, and the number of equations is equal to the number of variables in the system. Inconsistent systems have no solution, and dependent systems have infinitely many solutions.
Some common mistakes when solving a system of equations include making errors in algebraic manipulation, not checking the solution in all equations, and misinterpreting the solution as a coordinate instead of the values of the variables. It is important to double-check your work and always verify the solution in all equations.