Gaussian Jordan Method in solving in Solving System of Equations
(read the previous posts so you'll not be lost)

Tip:
dodge making the rows with fraction values
you can eliminate them by not just only multiplying a scalar.. you can add a multiple of other equation to another equatiion

Steps:
Put system of Equation to matrix form
You need to put the matrix to echelon form using row operations
you must leave each row only one variable ( diff variables each row)
after that put to system and write coordinate form of the solution

(1,4,-2)

matrix
a rectangular array of number
its size is row x column or m x n
-------> this is a 2 x 3 matrix
matix with same number of columns and rows is called square matrix

2 types of matrix coefficient and augmented
coefficient matrix-- matrix which only the coefficient of the variables in the given system is written
x+9y-13z=2
20x+5y+-6z=9

augmented matrix-- matrix which the constant is written
x+9y-13z=2
20x+5y+-6z=9

Elementary  Row Operations
used in system of equation to solve for solutions
1. Interchange two rows (or columns).
2. Multiply each element in a row (or column) by a non-zero number.
3. Multiply a row (or column) by a non-zero number and add the result to another row (or column).

EcHELON Form
form of system of equation in which the a variable in the first is not included in the second up to the last and the variables of the first variable in each equation is one.
 to get echelon arrange the variables first and use elementary  row operatiions to get the echelon form..


ex.of echelon form
x+3y+4z=22
      y+3z=12
           z=3

after that use back subtitution to find the solutions.. substitute the value of the variable on the third equation in the echelon form and substitute the corresponding values to the first to find the variable leading in the first equation.
so z=3 then y+3(3)= 12 then y= 3 then substitute to the first x+3(3)+4(3)=22 then x= 1
so the solution is in this form (x,y,z) then (1,3,3) is the solution..

other way to solve system of equations

Inconsistent -----------------------> parallel LINES

this is when a1/a2 = b1/b2 and not equal to c1/c2

when the coefficient in the variables are the same but different constant

http://www.algebra.com/algebra/homework/coordinate/Types-of-systems-inconsistent-dependent-independent.lesson  <---- for more info

This lesson concerns systems of two equations, such as: 

2x + y = 10

3x + y = 13.

The equations can be viewed algebraically or graphically. Usually, the problem is to find a solution for x and y that satisfies both equations simultaneously. Graphically, this represents a point where the lines cross. There are 3 possible outcomes to this (shown here in blue, green, and red):

The two lines might not cross at all, as in 


y = x
y = x + 10.

This means there are no solutions, and the system is called inconsistent. 

If you try to solve this system algebraically, you'll end up with something that's not true, such as 0 = 10. 

Whenever you end up with something that's not true, the system is inconsistent.

consistent(dependent)---> coinsiding lines

If the equations on a give equation have they only have one 

solution

a1/a2 is  equal to b1/b2 is  equal to c1/c2<<< construction of the 

coefficient of the variables in the equations

there are infinite solutions... but you need to put parameters

The graphical solution is any point on the two identical straight lines.

to find the solution of the given system you need to find the value interms of x and y
and replace them with other variable.. like  the solution of given system  : (1-b,b) which y=t... it is most ad visable to change the variable of the rightmost in the equation...

intersecting lines are consistent(independent)??

consistent(independent)---> intersecting lines

If the equations on a give equation have they only have one solution
a1/a2 is not equal to b1/b2 is not equal to c1/c2<<< construction of the coefficient of the variables in the equations

Problem
Find all solutions to the system y – x = 1 and y + x = 3.

First, graph both equations on the same axes.
 The two lines intersect once. That means there is only one solution to the system.

The point of intersection appears to be (1, 2).
Read the point from the graph as accurately as possible.

y – x = 1
2 – 1 = 1
1 = 1

TRUE

(1, 2) is a solution of y – x = 1.
y + x = 3
2 + 1 = 3
3 = 3

TRUE

(1, 2) is a solution of y + x = 3.
Check the values in both equations. Substitute 1 for x and 2 for y. (1, 2) is a solution.
Answer
(1, 2) is the solution to the system y – x = 1 and
y + x = 3.
Since (1, 2) is a solution for each of the equations in the system, it is the solution for the system.

what are you?

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It is the study of lines, planes, and subspaces and their intersections using algebra. Linear algebra assigns vectors as the coordinates of points in a space, so that operations on the vectors define operations on the points in the space. <<< from wikipedia

Linear algebra is the study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of rotations in space,least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering. Confusingly, linear algebra is not actually an algebra in the technical sense of the word "algebra" (i.e., a vector space V over a field F, and so on). <<<<from   http://mathworld.wolfram.com/

linear algebra is studiies about linees.