INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS I

LINEAR EQUATION IN N VARIABLES

A linear equation in n variables x₁, x₂, x₃, … x_n has the form :
a₁x₁ + a₂x₂ + a₃x₃ + … + a_nx_n = b

The Coefficients a₁, a₂, a₃, … a_n are real numbers
The Constant Term b is a real number
The Leading Coefficient is the number a₁
The Leading Variable is x₁

EXAMPLES OF LINEAR EQUATION
A.) 3x + 2y = 7
B.) 5x + y = 4
C.) 10a + 3b + 4c = 12

A Solution of a linear equation in n variables is a sequence of n real numbers s₁, s₂, s₃, … ,s_n arranged so the equation is satisfied when the values

 x₁ = s₁ , x₂ = s₂, x₃ = s₃, . . . , x_n = s_nare substituted into the equation
 

 The Solution set is the set of all solution of a linear equation, to describe the entire solution set of a linear equation. A Parametric Representation is often used

EXAMPLE OF A PARAMETRIC REPRESENTATION
Solve the Linear Equation: x + 2y = 4

* Solve for one of the variables on terms of the other variable
x = 4 – 2y

In this form, the variable y is Free, which means it can take on any real value
* Then represent the free variable as a Parameter by introducing a third variable

Let x₂ = v, V is an element all real numbers
x₁ = 4 – 2v
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