If you have a system fo linear equations, the solution is where the equations intersect
Using row reduction you get a system of linear equations which still satisfies the intersection
Why are free variables used?
The values in which the free variable can be are limited in a range as for if the solution is a line and not a plane
So what is the point in writing the solutions as a set of vector additions using free variables?
If z in this case is a free variable, writing the solution in a vector equation
Where , denotes a new row
Solving for the pivot variables x and y and writing into decomposed vector form
[x,y] = [4,0] - [0,3] + z[1,2]
Z is said to be any real number zeR But looking at the graph, when z = 4, y=11/8 is not on the line of intersection So why do they say that z is a free variable when it isnt?