Linear Equations: Chapter 1.1

Chapitre 1: Systèmes d'Équations Linéaires

1.1 Introduction

• A linear equation with n variables $x_1, x_2,..., x_n$ has the form $a_1x_1 + a_2x_2 +... + a_nx_n = b$, where $a_1, a_2,..., a_n$ and $b$ are real constants.
• A solution to a linear equation is a list of n numbers $s_1, s_2,..., s_n$ that satisfy the equation when substituted for $x_1, x_2,..., x_n$ respectively.
• The set of all solutions to a linear equation is the solution set of the equation.
• A system of linear equations is a finite set of linear equations.
• A solution to a system of linear equations is a list of numbers that solves each equation in the system.
• The set of all solutions to a system of linear equations is the solution set of the system.
• Solving a system of linear equations means finding all of its solutions.
• Example: $x + y = 3$ is a linear equation in two variables and $x_1 - 2x_2 + 3x_3 = 0$ is a linear equation in three variables.
• Non-Linear Examples*
• $x_1 + x_2 = x_1x_2$ , $\sqrt{x} + y = 5$, and $x + sin(y) = 0$ are not linear equations.
• A system of linear equations is considered compatible if it possesses at least one solution.
• A system of linear equations is incompatible if it lacks any solutions.

1.2 Systems of two linear equations with two unknowns

• The general system of two linear equations with two unknowns is: $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$, where $a_1, b_1, c_1, a_2, b_2, c_2$ are real constants.
• Geometric Interpretation*
• Each equation corresponds to a line in the plane.
• A solution to the system is a point $(x, y)$ on both lines.
• Three Possible Cases*
• Lines intersect at one point: system has a unique solution.
• Lines are parallel and distinct: system has no solution.
• Lines are identical: system has infinitely many solutions, representing the line itself.
• Resolution is achieved via substitution, elimination, or matrices.

1.3 Systems of linear equations: Matrix notation

• The coefficient matrix is formed by the coefficients of the variables in a system of linear equations.
• The augmented matrix is formed by adding a column of constants from the right side of the equations to the coefficient matrix.
• Example*
• For the system:
  • $2x + 3y - z = 1$
  • $x - y + 2z = 3$
  • $3x + 2y + z = 4$
• The coefficient matrix is:
  • $\begin{bmatrix} 2 & 3 & -1 \ 1 & -1 & 2 \ 3 & 2 & 1 \end{bmatrix}$
• The augmented matrix is:
  • $\begin{bmatrix} 2 & 3 & -1 & 1 \ 1 & -1 & 2 & 3 \ 3 & 2 & 1 & 4 \end{bmatrix}$