# Equation and system in N

• June 8th 2009, 12:54 PM
dhiab
Equation and system in N
• June 8th 2009, 01:16 PM
VonNemo19
What do you mean by $N^2$?
• June 8th 2009, 05:14 PM
Soroban
Hello, dhiab!

These are strange problems . . .

Quote:

Solve in $N^2\!:\quad\begin{array}{ccc}xy & \leq & 2x \\ x+y &=& 4\end{array}$

The line $x + y \;=\:4$ has intercepts (4,0) and (0,4).
Code:

|
* |
*
| *
|  *
|    *
- - + - - - * - -
|        *
|

We have the inequality: . $xy - 2x \:\leq\:0 \quad\Rightarrow\quad x(y-2) \:\leq\:0$

This is true when $x$ and $(y-2)$ have opposite signs.

There are two cases: . $\begin{Bmatrix}x \:\geq \:0 \\ y \:\leq \:2\end{Bmatrix}\;\text{ and }\;\begin{Bmatrix}x \:\leq\:0 \\ y \:\geq \:2\end{Bmatrix}$

The graph looks like this:
Code:

::::::::|
::::::::|
::::::::|
::::::::|
- - - - 2+ - - - - - -
|::::::::::
------------+:-:-:-:-:----
|::::::::::
|::::::::::
|::::::::::

Together, we have this graph . . .
Code:

::::*:::|
::::::*:|
::::::::*
::::::::| *
- - - - 2+ - * - - - -
|:::::*:::::::
----------+:-:-:-:*:-:-:-
|:::::::::*:::
|:::::::::::*:
|:::::::::::::*

The final graph looks like this:
Code:

*  |
* |
*
|
2+ - * - - - -
|    *
----------+-------*------
|        *
|          *
|            *

How can we describe this graph?

Maybe: . $x + y \:=\:4\:\text{ for }(x \leq 0) \cup (x \geq 2)$

• June 8th 2009, 09:32 PM
dhiab
Quote:

Originally Posted by VonNemo19
What do you mean by $N^2$?

http://www.mathramz.com/xyz/latexren...efd0b9ff1b.png = N ×N
• June 8th 2009, 10:50 PM
mr fantastic
Quote:

Originally Posted by dhiab

That is, the set of ordered pairs such that each element in the ordered pair is a natural number.
• June 9th 2009, 08:46 AM
dhiab
Quote:

Originally Posted by Soroban
Hello, dhiab!

These are strange problems . . .

The line $x + y \;=\:4$ has intercepts (4,0) and (0,4).
Code:

|
* |
*
| *
|  *
|    *
- - + - - - * - -
|        *
|

We have the inequality: . $xy - 2x \:\leq\:0 \quad\Rightarrow\quad x(y-2) \:\leq\:0$

This is true when $x$ and $(y-2)$ have opposite signs.

There are two cases: . $\begin{Bmatrix}x \:\geq \:0 \\ y \:\leq \:2\end{Bmatrix}\;\text{ and }\;\begin{Bmatrix}x \:\leq\:0 \\ y \:\geq \:2\end{Bmatrix}$

The graph looks like this:
Code:

::::::::|
::::::::|
::::::::|
::::::::|
- - - - 2+ - - - - - -
|::::::::::
------------+:-:-:-:-:----
|::::::::::
|::::::::::
|::::::::::

Together, we have this graph . . .
Code:

::::*:::|
::::::*:|
::::::::*
::::::::| *
- - - - 2+ - * - - - -
|:::::*:::::::
----------+:-:-:-:*:-:-:-
|:::::::::*:::
|:::::::::::*:
|:::::::::::::*

The final graph looks like this:
Code:

*  |
* |
*
|
2+ - * - - - -
|    *
----------+-------*------
|        *
|          *
|            *

How can we describe this graph?

Maybe: . $x + y \:=\:4\:\text{ for }(x \leq 0) \cup (x \geq 2)$

Hello : 0n has two cases in question 2 : x=0 and x no zero