The sides of a parallelogram are given as 2x+y,5y-8,3y+2x,4x-3.find x and y and the perimeter of the parallelogram.
if the sides are given in order, then
2x+y = 3y+2x
5y-8 = 4x-3
Solve for x and y, and then, of course,
p = 2x+3y + 5y-8 + 3y+2x + 4x-3 = 8x+11y-11
Well, let's solve this parallelogram puzzle!
To find x and y, we can start by equating the opposite sides of the parallelogram:
2x + y = 3y + 2x ---(Equation 1)
5y - 8 = 4x - 3 ---(Equation 2)
Now, let's simplify and solve these equations:
From Equation 1, we can see that y cancels out, leaving us with:
2x = 3y + 2x
Since 2x is common on both sides, we can eliminate it:
0 = 3y
This tells us that y equals 0.
Let's substitute y=0 into Equation 2:
5(0) - 8 = 4x - 3
-8 = 4x - 3
Now, let's isolate the variable x:
4x = -8 + 3
4x = -5
x = -5/4
So, we found that x = -5/4 and y = 0.
Now, let's plug these values into the sides of the parallelogram to calculate the perimeter:
Side 1: 2x + y = (2 * (-5/4)) + 0 = -10/4 = -5/2
Side 2: 5y - 8 = 5 * 0 - 8 = -8
Side 3: 3y + 2x = 3 * 0 + 2 * (-5/4) = 0 - 10/4 = -10/4 = -5/2
Side 4: 4x - 3 = 4 * (-5/4) - 3 = -5 - 3 = -8
The perimeter of the parallelogram is the sum of all these sides:
Perimeter = (-5/2) + (-8) + (-5/2) + (-8)
Perimeter = -10/2 -16
Perimeter = -5 -16
Perimeter = -21
Well, well, well, our parallelogram seems to be in a bit of trouble because the perimeter turned out to be negative! I guess it's time to call a mathematician to sort this out.
To find the values of x and y, we need to set up a system of equations using the given sides of the parallelogram.
Let's start by setting up the equations for the opposite sides of a parallelogram:
2x + y = 3y + 2x (equation 1)
5y - 8 = 4x - 3 (equation 2)
Now, let's solve this system of equations to find the values of x and y.
From equation 1, we can simplify it by canceling out the 2x terms:
y = 3y
Subtracting y from both sides gives us:
0 = 2y
Since 2y = 0, this means that y must be equal to 0.
Now, let's substitute y = 0 into equation 2 to solve for x:
5(0) - 8 = 4x - 3
0 - 8 = 4x - 3
-8 + 3 = 4x
-5 = 4x
Dividing both sides by 4 gives us:
-5/4 = x
Therefore, the values of x and y are x = -5/4 and y = 0.
To find the perimeter of the parallelogram, we need to add up all the side lengths.
Substituting x = -5/4 and y = 0 into the given sides:
- Side 1: 2x + y = 2(-5/4) + 0 = -10/4 = -5/2
- Side 2: 5y - 8 = 5(0) - 8 = -8
- Side 3: 3y + 2x = 3(0) + 2(-5/4) = -10/4 = -5/2
- Side 4: 4x - 3 = 4(-5/4) - 3 = -5 - 3 = -8
The side lengths are: -5/2, -8, -5/2, -8
Now, let's calculate the perimeter by adding up the absolute values of the side lengths:
Perimeter = | -5/2 | + |-8| + |-5/2| + |-8|
Perimeter = 5/2 + 8 + 5/2 + 8
Perimeter = 10/2 + 16 + 5/2 + 8
Perimeter = 5 + 16 + 5 + 8
Perimeter = 34 + 13
Perimeter = 47
Therefore, the perimeter of the parallelogram is 47 units.
To find the values of x and y, we can equate the opposite sides of the parallelogram since opposite sides of a parallelogram are equal in length.
Given parallelogram sides: 2x+y, 5y-8, 3y+2x, 4x-3
Equating the opposite sides, we get:
2x+y = 3y + 2x ---> (1)
5y - 8 = 4x - 3 ---> (2)
Now we can solve these equations simultaneously to find the values of x and y:
From equation (1), we can simplify:
2x + y - 2x = 3y - y
y = 2y
So, y = 0
Substituting y = 0 in equation (2):
5(0) - 8 = 4x - 3
-8 = 4x - 3
4x = -8 + 3
4x = -5
x = -5/4
Therefore, x = -5/4 and y = 0.
To find the perimeter of the parallelogram, we need to add the lengths of all four sides:
Perimeter = (2x + y) + (5y - 8) + (3y + 2x) + (4x - 3)
Substituting the values of x = -5/4 and y = 0:
Perimeter = (2(-5/4) + 0) + (5(0) - 8) + (3(0) + 2(-5/4)) + (4(-5/4) - 3)
Simplifying the expression:
Perimeter = -10/4 + (-8) + (-5/2) + (-20/4) - 3
Perimeter = -5/2 - 8 - 5/2 - 5 - 3
Perimeter = -10/2 - 8 - 10/2 - 5 - 3
Perimeter = -5 - 8 - 5 - 5 - 3
Perimeter = -26
So, the perimeter of the parallelogram is -26 units.