Question

How do we use simultaneous equations for these two questions ?

Answer 1

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!

Answer 2

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`]