# Need quick help with equation

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• Nov 7th 2006, 07:30 AM
Neurok
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.

Please someone help.
• Nov 7th 2006, 07:49 AM
topsquark
Quote:

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
• Nov 7th 2006, 07:54 AM
topsquark
Quote:

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
• Nov 7th 2006, 08:16 AM
Neurok
Thank you.