 # Solving Systems of Linear Equations

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

• how to solve a system of linear equations in two variables
• how to solve a system of linear equations in three variables with solution
• how to solve a system of linear equations in three variables with no solution or infinite solutions

### Solving a System of Linear Equations in Two Variables

A system of linear equations is two or more equations that contain the same variables. A solutions to a system of equations are the point where the lines intersect. There are four methods to solving systems of linear equations: graphing, substitution, elimination and matrices. Solving systems of equations first shows up in Algebra I, but more complex applications occur in Algebra II.

Solving Linear Systems of Equations Using Substitution
3 complete examples are shown along with an outline of the basic idea
Solving Systems of Linear Equations Using Elimination By Addition - Two complete examples and part of a third problem are shown

### Solving a Linear System in Three Variables with a Solution

In Algebra II, sometimes we will be asked to solve systems of equations three variables. When solving these systems of equations, a 3D coordinate system is necessary since systems of equations with three variables are not linear. Therefore, solving these systems of equations by graphing is not possible. Solving by substitution would be difficult, so we often solve by addition and elimination.

This video explains how to solve a system of equations in three variables and shows the 3 possible results graphically. Solving a System of Equations Involving 3 Variables
Using Elimination by Addition - Example 1. Solving a System of Equations Involving 3 Variables
Using Elimination by Addition - Example 2. Solving a System of Equations Involving 3 Variables
Using Elimination by Addition - Example 3

### Solving a Linear System in Three Variables with no or Infinite Solutions

Sometimes we have a system of equations that has either infinite or zero solutions. We call these no solution systems of equations. When we solve a system of equations and arrive at a false statement, it tells us that the equations do not intersect at a common point. One scenario is that 2 or more of the planes are parallel or that two of the planes intersect and the other intersects at a different point

This video provides examples of a system with no solution and infinite solutions. The answers are verified graphically. Solving systems with 3 variables: cases of infinite solutions and no solutions

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