Unit 6: Systems of Linear Equations and Inequalities
In previous units, we learned that linear equations with one variable generally have one solution. However, linear equations with two variables have an infinite number of solutions. If we pair two linear equations together, we can solve for the pair of numbers that would solve both equations. This is called a system of linear equations. In this unit, we will learn how to solve systems of linear equations.
Completing this unit should take you approximately 3 hours.
6.1: Solution of a System of Linear Equations
First, we'll look at how to determine whether a given ordered pair of numbers is a solution to a system of linear equations. We do this using the techniques you already learned in this course.
6.2: Solving Systems of Linear Equations by Graphing
There are several methods for solving systems of linear equations. The first method we explore is using graphing to solve systems of linear equations.
6.3: Solving Systems of Linear Equations Using the Substitution Method
Sometimes it is not practical to solve a system of equations by graphing. The substitution method is a non-graphical method for solving systems of equations, and it uses algebra techniques you already know.
For the substitution method, we solve one of the equations for one of the variables in terms of the other variable. Then, we substitute that expression into the second equation to solve for the variable. Then, once you have solved for one variable, you can solve for the other.
6.4: Solving Systems of Linear Equations Using the Elimination Method
The elimination method is another non-graphical method for solving systems of linear equations. For this method, we use the addition property of equality. This states that if we add the same amount to both sides of an equation, the two sides of the equation will still be equal.
6.5: Choosing a Strategy for Solving Systems of Linear Equations
It can be difficult to decide which of the three methods (graphing, substitution, or elimination) is best for solving a given system of linear equations.
6.6: Solving Word Problems by Using Systems of Equations
Linear systems of equations appear in many real-world applications. Systems of equations are often used to conduct comparisons, such as when comparing prices. It can be difficult to translate a world problem into equations and to determine the appropriate variables.
6.7: Graphing Systems of Linear Inequalities
We can use graphical methods to solve systems of linear inequalities, much like we graphed single inequalities earlier in this course.
6.8: Applications of Systems of Linear Inequalities
Linear inequalities appear in many real-world applications.