Mathematics

Systems of Linear Equations


1 Introduction to Systems of Linear Equations
2 Gaussian Elimination and Gauss-Jordan Elimination

Geetha Sivaraman
St. Joseph’s College Tiruchirappalli

, 1.1 Introduction to Systems of Linear Equations

◼ a linear equation in n variables:


a1,a2,a3,…,an, b: real number
a1: leading coefficient
x1: leading variable
◼ Notes:
(1) Linear equations have no products or roots of variables and
no variables involved in trigonometric, exponential, or
logarithmic functions.

(2) Variables appear only to the first power.

, ◼ Ex 1: (Linear or Nonlinear)
1
Linear (a ) 3x + 2 y = 7 ( b) x + y −  z = 2 Linear
2


Linear ( c ) x1 − 2 x 2 + 10 x3 + x 4 = 0 ( d ) ( sin ) x1 − 4 x 2 = e 2
Linear
2
Exponentia l
Nonlinear ( e ) xy + z = 2 ( f ) e x
− 2y = 4 Nonlinear
not the first power
1 1
Nonlinear ( g ) sinx1 + 2 x2 − 3x3 = 0 (h) + = 4 Nonlinear
x y
trigonomet ric functions not the first power

, ◼ a solution of a linear equation in n variables:
a1 x1 + a2 x2 + a3 x3 +  + an xn = b

x1 = s1 , x2 = s2 , x3 = s3 , , xn = sn
such a1s1 + a2 s2 + a3 s3 +  + an sn = b
that
◼ Solution set:
the set of all solutions of a linear equation