Solve one of the equations for L, then substitute that value into the other equation to solve for W. Then take that value of W and substitute it into either equation to solve for L.
Let's start with (eq2):
18 = L * W
18 / W = L
Now substitute L = 18 / W into (eq1):
18 = 2L + 2W
18 = 2(18/W) + 2W
18 = 36/W + 2W
Multiply both sides by W to get rid of the fractions:
18W = 36 + 2W^2
Divide both sides by 2 to get:
9W = 18 + W^2
Subtract 9W from both sides to get:
0 = W^2 - 9W + 18
Which can be written as:
W^2 - 9W + 18 = 0
Which factors into:
(W - 3)(W - 6) = 0
That means W = 3 or W = 6.
Take W = 3 and plug it into (eq2):
18 = L * W
18 = L * 3
18 / 3 = L * 3 / 3
6 = L
Take W = 6 and plug it into (eq2):
18 = L * W
18 = L * 6
18 / 6 = L * 6 / 6
3 = L
So either L = 3 and W = 6 or L = 6 and W = 3.
Either way, we can check our work by plugging both values into (eq1) and (eq2):
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Verified answer
Let L = length
Let W = width
The perimeter of a rectangle is:
p = length + width + length + width
p = 2L + 2W
18 = 2L + 2W
Area of a rectangle = length * width
a = L * W
18 = L * W
So we have two equations and two unknowns:
(eq1): 18 = 2L + 2W
(eq2): 18 = L * W
Solve one of the equations for L, then substitute that value into the other equation to solve for W. Then take that value of W and substitute it into either equation to solve for L.
Let's start with (eq2):
18 = L * W
18 / W = L
Now substitute L = 18 / W into (eq1):
18 = 2L + 2W
18 = 2(18/W) + 2W
18 = 36/W + 2W
Multiply both sides by W to get rid of the fractions:
18W = 36 + 2W^2
Divide both sides by 2 to get:
9W = 18 + W^2
Subtract 9W from both sides to get:
0 = W^2 - 9W + 18
Which can be written as:
W^2 - 9W + 18 = 0
Which factors into:
(W - 3)(W - 6) = 0
That means W = 3 or W = 6.
Take W = 3 and plug it into (eq2):
18 = L * W
18 = L * 3
18 / 3 = L * 3 / 3
6 = L
Take W = 6 and plug it into (eq2):
18 = L * W
18 = L * 6
18 / 6 = L * 6 / 6
3 = L
So either L = 3 and W = 6 or L = 6 and W = 3.
Either way, we can check our work by plugging both values into (eq1) and (eq2):
(eq1): 18 = 2L + 2W
18 = 2(3) + 2(6)
18 = 6 + 12
18 = 18, so it checks out!
(eq1): 18 = 2L + 2W
18 = 2(6) + 2(3)
18 = 12 + 6
18 = 18, so it checks out!
(eq2): 18 = L * W
18 = 3 * 6
18 = 18, so it check sout!
(eq2): 18 = L * W
18 = 6 * 3
18 = 18, so it check out!
Let breadth = a
Let length = b
Perimeter
a + a + b + b = 18
2a + 2b = 18
a + b = 9
Area
a * b = 18
a = 18 / b
Sub into Perimeter...
18/b + b = 9
b^2 - 9b + 18 = 0
(b - 6) (b - 3) = 0
b = 6 or b = 3
a = 18/6 or a = 18/3
a = 3 or a = 6
Therefore, one side of the rectangle is 3 and the other side is 6 :)
How about just guessing: find 2 numbers which sum up to 9 and multiplied to 18.
hummm, can it be, (1,18)? (2,9)? or (3,6) yesssss.
I now know for sure that it is (3,6) but I can never figure out which is the length and which is the width........
Factoring.
Good luck with that.