Question

Please can you help me find x and y in the picture below

Answer 1

Explanation:

Step 1:

Length l = x cm,

Width w = (x+y) cm,

Area = 68 cm2,

So Area = L * W

68 = x(x+y)

Step 2

Perimeter = 2(l+w) (since it is a 4-sided rectangle)

42 = 2(x+(x+y))

21 = 2x+y

y = 21-2x

Step 3

plug this back into the first equation we had

68 = x(x+21-2x)

68 = x(x+21-2x)

68=x(21-x)

Once we simplify this we should be able to get

68=21x-
`x^(2)`

Rewrite this as

`x^(2) -21x+68`

Step 4

Factor this and get (x-4)(x-17)

so x = 4 and x = 17 by setting both of these equal to 0

x-4=0

x=4

x-17=0

x=17

Step 5

Plug these back into your y equation

for x=17

y=21-2(17)

y=21-34

y=-13 (negative values are not possible so this cannot be the answer)

x=4

y=21-2(4)

y=21-8

y=13

Therefore the dimensions of the rectangle have to be in cm ; x=4 and y=13

Answer 2

x = 4

y = 13

Explanation:

Given that a rectangle has side lengths of x cm and (x + y) cm, where x and y are both positive, then:

`\textsf{Width:}\quad w = x \;\sf cm`

`\textsf{Length:}\quad l = (x + y) \;\sf cm`

The perimeter of a rectangle is twice the sum of its width and length.

Given that the perimeter of the rectangle is 42 cm, then:

`\begin{aligned}\sf Perimeter&=2(w + l)\\42&=2(x+(x+y))\\42&=2(2x+y)\\21&=2x+y\end{aligned}`

Rearrange the equation to isolate y:

`y=21-2x`

The area of a rectangle is the product of its width and length.

Given that the area of the rectangle is 68 cm², then:

`\begin{aligned}\sf Area&=w * l\\68&=x * (x+y)\\68&=x^2+xy\end{aligned}`

Substitute the expression for y into the equation for area:

`\begin{aligned}68&=x^2+x(21-2x)\\68&=x^2+21x-2x^2\\68&=21x-x^2\\x^2-21x+68&=0\end{aligned}`

Factor the quadratic equation and solve for x:

`\begin{aligned}x^2-21x+68&=0\\x^2-17x-4x+68&=0\\x(x-17)-4(x-17)&=0\\(x-4)(x-17)&=0\\\\x-4&=0 \implies x=4\\x-17&=0 \implies x=17\end{aligned}`

Substitute the values of x into the equation for y:

`\begin{aligned}x=4 \implies y&=21-2(4)\\y&=21-8\\y&=13\end{aligned}`

`\begin{aligned}x=17 \implies y&=21-2(17)\\y&=21-34\\y&=-13\end{aligned}`

As x and y are positive, the values of x and y are:

• 
  `x = 4`
• 
  `y = 13`