# Thread: Need quick help with equation

1. ## Need quick help with equation

1.Two one by one natural numbers sum 109 is smaller then there product.Find those numbers.
{y=x+1
xy=x+y+109

x=y-1
(y-1)y=x+y+109 Whats next?

2. Right-angled parterres area is 56m2(square),and the fence,that surrounds the parterre is 30m.Find the parterres dimension.

2. Originally Posted by Neurok
1.Two one by one natural numbers sum 109 is smaller then there product.Find those numbers.
{y=x+1
xy=x+y+109

x=y-1
(y-1)y=x+y+109 Whats next?
I'm going to assume you set up the problem correctly because I can't make sense out of the question.

First you need to replace ALL x's in the second equation:
$\displaystyle xy = x + y + 109$

$\displaystyle (y - 1)y = (y - 1) + y + 109$

$\displaystyle y^2 - y = 2y + 108$

$\displaystyle y^2 - 3y - 108 = 0$

$\displaystyle (y - 12)(y + 9) = 0$

Thus y = 12 or y = -9. This means that x = 11 or x = -10 respectively.

So your solutions are: (x, y) = (11, 12) or (x, y) = (-10, -9).

-Dan

3. Originally Posted by Neurok
2. Right-angled parterres area is 56m2(square),and the fence,that surrounds the parterre is 30m.Find the parterres dimension.
I'm assuming your parterre is a rectangle, rather than a square. So let the dimensions of the parterre be x and y. The we know that
$\displaystyle xy = 56$
$\displaystyle 2x + 2y = 30$

From the second equation:
$\displaystyle 2x + 2y = 30$

$\displaystyle x + y = 15$

$\displaystyle y = 15 - x$

Inserting this into the first equation gives:
$\displaystyle xy = 56$

$\displaystyle x(15 - x) = 56$

$\displaystyle 15x - x^2 = 56$

$\displaystyle x^2 - 15x + 56 = 0$

$\displaystyle (x - 7)(x - 8) = 0$

So x = 7 m or x = 8 m. Thus y = 8 m or y = 7 m respectively. Either way the dimensions of the parterre are 7 m by 8 m.

-Dan

4. Thank you.