Algebra I: A Basic Course in Abstract Algebra

Chapter 10

System of Linear Equations

An equation of the type ax = b, where a and b are real constants and x is an unknown, is called a linear equation in x. If x₁, x₂ …, x_n are unknowns (called variables) then a relation of the type

 

a₁x₁ + a₂x₂ + : : : + a_nx_n = b      (10.1)

 

where a₁, …, a_n and b are given real constants is called a linear equation in x₁,… , x_n. The constants a₁, …, a_n are called the coefficients of the variables x₁, …, x_n respectively. By a solution of equation (10.1) is meant a set of values of x₁, …, x_n which satisfy it.

Example 10.1.

1. 3x–0.2y = 5 is a linear equation in two variables x and y. x = 1, y = –10 is a solution. More solutions also exist.
2. is a linear equation in three variables x, y, z. x = 0, y = 3, z = 0 ...