Solving a System of Equations – Introducing the Basics

A system of equations is a set of multiple equations, which are all related to each other. When solving a system of equations, the goal is to find variables that will make each equation in the system true. It is possible to solve a system of equations with one, two, or more equations. The most common method of solving a system of equations is by substituting and/or graphing.

In order to effectively solve a system of equations, it is important to understand the structure of a system, as well as the different types of systems and how to solve them. A system of equations can be written in terms of the variables in the system, and the solution to the system consists of the values of the variables that make all of the equations true.

Substitution Method

The substitution method is one of the most common methods for solving a system of equations. In this method, one variable is chosen, and its value is computed from one of the equations. This variable is then substituted into the other equations to obtain a solution. The procedure for using the substitution method is as follows:

1. Choose one equation and identify two variables.
2. Solve for one of the variables in terms of the other.
3. Substitute the expression for one variable into the other equations of the system.
4. Solve the resulting equation for the remaining variable.
5. Substitute both variables into the original equations to verify that a solution has been found.

Graphing Method

Another popular method of solving a system of equations is the graphing method. This method involves graphing the equations of the system on the same coordinate plane to produce an elongated shape, also known as a system of curves. The solution of the system is then determined by finding the point of intersection of the curves. The procedure for solving a system of equations by graphing is as follows:

1. Graph the equations of the system on the same coordinate plane.
2. Locate the point of intersection of the curves.
3. Substitute the point of intersection into the equations to verify that a solution has been found.

Summary

Solving a system of equations relies on understanding the structure of the system and the different methods for solving systems of equations. The two most common methods for solving a system of equations are the substitution method and the graphing method. The substitution method requires selecting one equation and finding an expression for one variable in terms of the other. The graphing method consists of graphing the equations of the system on the same coordinate plane and then finding the point of intersection of the curves. In both cases, the solutions are then verified by substituting the variables into the original equations.