Skip to main content

Chapter2Systems of Linear Equations: Algebra

Primary Goal

Solve a system of linear equations algebraically in parametric form.

This chapter is devoted to the algebraic study of systems of linear equations and their solutions. We will learn a systematic way of solving equations of the form

ADCDB 3 x 1 + 4 x 2 + 10 x 3 + 19 x 4 − 2 x 5 − 3 x 6 = 1417 x 1 + 2 x 2 − 13 x 3 − 7 x 4 + 21 x 5 + 8 x 6 = 2567 − x 1 + 9 x 2 + 3 2 x 3 + x 4 + 14 x 5 + 27 x 6 = 26 1 2 x 1 + 4 x 2 + 10 x 3 + 11 x 4 + 2 x 5 + x 6 = − 15.

In Section 2.1, we will introduce systems of linear equations, the class of equations whose study forms the subject of linear algebra. In Section 2.2, will present a procedure, called row reduction, for finding all solutions of a system of linear equations. In Section 2.3, you will see how to express all solutions of a system of linear equations in a unique way using the parametric form of the general solution. Finally, in Section 2.4, we will learn how to write a system of linear equations succinctly as a matrix equation, which looks like Ax = b , where A is an m × n matrix, b is a vector in R m and x is a variable vector in R n .