6: Systems of Equations and Inequalities
In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions.
-
- 6.1: Prelude to Systems of Equations and Inequalities
- In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.
-
- 6.2: Systems of Linear Equations - Two Variables
- A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution.
-
- 6.3: Systems of Linear Equations with Three Variables
- A solution set is an ordered triple that represents the intersection of three planes in space. A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. Systems of three equations in three variables are useful for solving real-world problems.
-
- 6.5: Solving Systems with Gaussian Elimination
- A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
Thumbnail: Possible types of solutions for the points of Intersection of a circle and an ellipse.
Contributors
-
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .