Simultaneous equations

[lovely_nancy]

could someone please help me with the parts of this question
http://i51.tinypic.com/11rg9ao.jpg

 

[Aladdin]

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For the first part :
It tells to find the total area,which is the area of the rectangle.

So Area of a rectangle = length*width = L*w

L=(x + y )+ ( x + y ) + x

w= (x + y) + x

By replacing you'll get the Area ...

 

[lovely_nancy]

then how do we show that the total area of the dividing wood is 7xy + 2y(squared)

 

[soroban]

Hello, lovely_nancy!

\(\displaystyle \text{A window consists of six square panes of glass }(x\text{ by }x\text{ meters})\)
. . \(\displaystyle \text{and all the wooden dividers are }y\text{ meters wide.}\)

    - *-------*---*-------*---*-------*
      |       |:::|       |:::|       |
    x |       |:::|       |:::|       |
      |       |:::|       |:::|       |
    - *-------*:::*-------*:::*-------*
    y |:::::::::::::::::::::::::::::::|
   -  *-------*:::*-------*:::*-------*
      |       |:::|       |:::|       |
    x |       |:::|       |:::|       |
      |       |:::|       |:::|       |
    - *-------*---*-------*---*-------*
      :   x   : y :   x   : y :   x   :

\(\displaystyle \text{(a) Write the area of the whole window in terms of }x\text{ and }y.\)

\(\displaystyle \text{The width of the window is: }\:3x+2y\,\text{ meters.}\)
\(\displaystyle \text{The height of the window is: }\:2x+y\,\text{ meters.}\)

\(\displaystyle \text{The area is: }\;(3x+2y)(2x+y) \;=\;6x^2 + 7xy + 2y^2\,\text{ square meters}\)

\(\displaystyle \text{ (b) Show that the area of the dividers is }\,7xy + 2y^2\)

\(\displaystyle \text{The area of the whole window is: }\,6x^2 + 7xy + 2y^2\)
\(\displaystyle \text{The area of the 6 panes of glass is: }\,6x^2\)

\(\displaystyle \text{Therefore, the area of the dividers is: }\x^2 + 7xy + 2y^2) - 6x^2 \;=\;7xy + 2y^2\)

\(\displaystyle \text{The total area of glass is 1.5 m}^2.\)
\(\displaystyle \text{The total area of the dividers is 1 m}^2.\)

\(\displaystyle \text{Find }x\text{, and find an equation for }y\text{, and solve.}\)

\(\displaystyle \text{The total area of glass is 1.5 m}^2: \;\;6x^2 \:=\:1.5 \quad\Rightarrow\quad x^2 \:=\:\tfrac{1}{4} \quad\Rightarrow\quad \boxed{x \:=\:\tfrac{1}{2}\text{ meter}}\)


\(\displaystyle \text{The total area of the dividers is 1 m: }\;7xy + 2y^2 \:=\:1\)

\(\displaystyle \text{Then we have: }\;7\left(\frac{1}{2}\right)y + 2y^2 \:=\:1 \quad\Rightarrow\quad 4y^2 + 7y - 2 \:=\:0\)

. . . \(\displaystyle (4y-1)(y+2) \:=\:0 \quad\Rightarrow\quad y \:=\:\tfrac{1}{4},\;-2\)

\(\displaystyle \text{Therefore: }\;\boxed{y \:=\:\tfrac{1}{4}\text{ meter}}\)