Systems of equations and inequalities

Many problems have more than one unknown, and more than one condition. For example: how many adult and child tickets were sold, if we know the total number of tickets and the total money? Each condition gives one equation. Together, the equations form a system.

In this chapter, you will learn several ways to solve systems of equations, including a method with matrices that works for any number of unknowns. At the end, you will graph systems of inequalities.

This chapter covers the following topics:

Systems of equations in two variables

Solving systems by substitution and by elimination, systems with no solution or infinitely many solutions, applications, and nonlinear systems.

Systems of linear equations in three variables

Triangular form, solving by elimination, inconsistent and dependent systems, and applications.

Matrices and Gaussian elimination

Augmented matrices, row operations, row-echelon form, and solving systems with Gaussian and Gauss-Jordan elimination.

Systems of inequalities

Graphing linear and nonlinear inequalities in two variables, systems of inequalities, and an introduction to linear programming.

Practice questions

Practice problems on every section of this chapter, with full step-by-step solutions.

Before you start

This chapter uses linear equations from Chapter 2, and lines and circles from Chapter 3.