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Systems of Equations

A System of Equations or a System of Linear Equations refers to a set of more than one multi-variable equations that are related to one another. These equations are related because the same variables appear across all the different equations involved in the set. Graphically, the solution(s) to the system is where the graphs intersect. Since the solution is true for all equations in the system, all of the graphs will intersect at that point. We illustrate this in the first two examples of the Substitution method.

In precalculus, we mostly deal with systems of equations involving either two or three variables.

Example of two-variable system of equations

where the two variables are x and y.

Example of three-variable system of equations

where the three variables are x, y and z.

In this section of Systems of Equations, we're going to learn different techniques for solving these equations.

There are a number of different methods that can be used to solve systems of equations including: Substitution, Elimination (also known as Gaussian Elimination), using Matrices (Row Echelon Elimination) and by Graphing the equations.

Intersection Formula for Two Linear Equations

A formula to solve for the intersection point of two lines can be given if we have two linear equations in slope-intercept form. Given two linear equations

The solution is given by

This coordinate is the point where both lines intersect.