# Thread: Equation and system in N

1. ## Equation and system in N

1 ) sOLVE in this equation : .

2 ) solve in the system :

3 ) solve in the system :

2. What do you mean by $N^2$?

3. Hello, dhiab!

These are strange problems . . .

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

4. Originally Posted by VonNemo19
What do you mean by $N^2$?
= N ŚN

5. Originally Posted by dhiab
= N ŚN
That is, the set of ordered pairs such that each element in the ordered pair is a natural number.

6. 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