# Finding the dimension of S

## Homework Statement

Let S denote (x,y,z) in R3 which satisfies the following inequalities:
-2x+y+z <= 4
x-2y+z <= 1
2x+2y-z <= 5
x >=1
y >=2
z >= 3

## Homework Equations

How to find the dimension of the set S ?

## The Attempt at a Solution

I have tried to transform the inequalities into matrix form but I'm not quite sure that this is even the right way.

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## Homework Statement

Let S denote (x,y,z) in R3 which satisfies the following inequalities:
-2x+y+z <= 4
x-2y+z <= 1
2x+2y-z <= 5
x >=1
y >=2
z >= 3

## Homework Equations

How to find the dimension of the set S ?

## The Attempt at a Solution

I have tried to transform the inequalities into matrix form but I'm not quite sure that this is even the right way.

When you want to approach such problems algebraically, it is more involved and complex than you might think at first. The first step is to eliminate all "algebraic" inequalities, leaving only simple bounds like x >= 1, etc. We do this by introducing so-called slack or surplus variables, one for each inequality. Thus. we re-write $-2x+y+z \leq 4$ as $-2x+y+z+s_1 = 4$, where $s_1 \geq 0$ is a slack variable. Similarly, $x-2y+z \leq 1$ becomes $x-2y+z+s_2 = 1$, where $s_2 \geq 0$ is another slack variable. Finally, we re-write $2x+2y-z \leq 5$ as $2x+2y-z+s_3 = 5$, where $s_3 \geq 0$ is still another slack variable. So, altogether your system becomes
$$\begin{array}[rcccc] -2x+y+z&+s_1& & & =4 \\ x-2y+x & &+s_2& &=1 \\ 2x+2y-z & & &+s_3&=5\\ \end{array}\\ x \geq 1, y \geq 2, z \geq 3, s_1, s_2, s_3 \geq 0$$