How do we use simultaneous equations for these two questions ?

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2 Answers
Aug 28, 2017

a) #x = 10# and #y = 8#
b) #A = 35#

Explanation:

The length of the region to plant the Rambutan trees is given by #x- 5#. The width is given by #15 - y#. The perimeter is therefore

#P = 2l + 2w = 2(x -5) + 2(15 - y) = 2x - 10 + 30 - 2y = 2x -2y+ 20#

Since we know the perimeter is #24# metres, we can set the equation to #24#.

#24 = 2x - 2y + 20#

Simplify to get:

#4 = 2x - 2y#

#2 = x - y#

#y = x -2#

Now we design the other equation. The area of the entire garden is given by #A = lw = 15x#. But we need to subtract the area of the Rambutan trees to find the area of the Durian trees.

#115 = A_"total" - A_"Rambutan"#

#115= 15x- ((x-5)(15 - y))#

#115 = 15x - (15x - 75 + 5y - xy)#

#115 = 75 - 5y + xy#

We can substitute the first equation into the second now.

#115 = 75- 5(x - 2) + x(x - 2)#

#115 = 75 - 5x + 10 + x^2 - 2x#

#115 = x^2 - 7x + 85#

#0= x^2- 7x - 30#

#0 = (x - 10)(x+ 3)#

#x = 10 or -3#

Since a negative answer is impossible in this scenario, we know that #x = 10#. This means that #y = 8#.

The area in metres of the region with the Rambutan trees is #(x - 5)(15 - y) = 5(7) = 35" m"^2#.

Hopefully this helps!

Aug 28, 2017

See below.

Explanation:

Calling #u,v# the sides of rambutan terrain we have

#{(y + u = 15), (x = v + 5), (x (y+u) - u v = 115), (2 (u + v) = 24):}#

and now solving for #u,v,x,y# we obtain

#{(x=10),(y=8),(u=7),(v=5):}#

and

#S_r = u v = 35# [#"m"^2#]