solving but substiution?

Question

a permimeter of a rectangle is 13cm and its area is 10cm^2, determine its length and width.

Answer 1

4, 5/2

2L+ 2W=13
LW = 10
W=10/L
2L + 20/L = 13
2L^2 - 13L + 20 = 0
(2L - 5)(L - 4) = 0
L = 4, W=5/2 or L = 5/2, W=4

Answer 2

Let x= the width
y= the length

The perimeter is equal to 2x+2y and is given as equal to 13cm. To express this as an equation we write:

2x+2y=13 equation 1)

The area is equal to xy and is given as equal to 10cm^2.
We also express this as an equation, thus:

xy=10 equation 2)


Now we have two unknowns and two equations. So we can solve for x and y. To use the method of substitution, we solve for x based on equation 1, thus:

2x=13-2y

Dividing both sides of the equation by 2, we get:

x=(13-2y)/2

Now we substitute this value of x in equation 2:

y(13-2y)/2=10

Simplify by multiplying the terms in brackets by y, and multiplying both sides of the equation by 2:

13y-2y^2=20

Now express this as a quadratic equation:

-2y^2+13y-20=0

Multiplying the equation by -1, we get:

2y^2-13y+20=0.

What are the factors of 20 and 2y^2 that will give us the middle term of -13y. This needs some trial and error:
(2y- )(y- ) will take care of 2y^2. (2y-5)(y-4) will take care of 20, and will give the middle term of -13y. If in your trial you write (2y-4)(y-5), you won't get -13y even if -4(-5) also equals 20.

Ok, so now the correct equation becomes :

(2y-5)(y-4)=0, meaning

2y-5=0
2y=5
y=5/2 and

y-4=0
y=4.

So y has two values, viz; 5/2 and 4.

Now solve for x by substituting in equation 1, the values of y:

2x+2(5/2)=13
2x+5=13
2x=13-5
2x=8
x=4

and

2x+2(4)=13
2x+8=13
2x=13-8
2x=5
x=5/2

Of course!

If y=5/2cm, x=4cm, and
If y=4cm, x=5/2cm.

Check your answer:

Perimeter of rectangle: 2(5/2)+2(4)=
5+ 8=13cm.

Area: 5/2(4)=10 cm^2

Answer 3

Do you mean solving BY substitution?
p = 2l + 2w
13 = 2l+2w
13-2l = 2w
(13-2l)/2 = w
a = l*w
10 = l*w
10 = l*(13-2l)/2
10 = (13l-2l^2)/2
20 = 13l - 2l^2
2l^2-13l+20 = 0
a=2
b=-13
c=20

l = (13+-sqrt(169-160))/4
l = (13+-3)/4
l = 10/4 or 4

(13-2(4))/2 = w
5/2 = w

(13-2(10/4))/2 = w
(13 - 5)/2 = w
4 = w
however 4 * 4 does not equal 10 so the real answers are
l = 4 and w = 5/2

Answer 4

yep thats right.

x = length y = hieght

10 = xy (area equation)

2x + 2y = 13 (perimeter equation)

so, solve this equation. solve the area equation for one variable and plug it into the perimeter equation