Solving for Rectangle Dimensions and Properties
The image shows a math problem related to a rectangle with the sides labeled as expressions in terms of x and y. To solve for x and y, we will use the properties of a rectangle, which state that opposite sides are equal. Therefore, we can set up the following equations based on the given expressions for each side:
1. The longer sides (length) of the rectangle are equal, so we have:
4x - y = 3x + 1
2. The shorter sides (width) of the rectangle are equal, so we have:
x + 6y = 2x + 2
Now we'll solve these two equations simultaneously.
From equation 1:
4x - y = 3x + 1
x - y = 1 (Subtract 3x from both sides)
From equation 2:
x + 6y = 2x + 2
6y = x + 2 (Subtract x from both sides)
Since from equation 1 we have x - y = 1, we can rearrange this to find the value of y:
y = x - 1 (Adding y to both sides and subtracting 1 from both sides)
Now we substitute the value of y back into the second equation to find x:
6y = x + 2
6(x - 1) = x + 2
6x - 6 = x + 2
6x - x = 2 + 6
5x = 8
x = 8 / 5
x = 1.6
Now, we substitute the value of x back into the equation y = x - 1 to find y:
y = 1.6 - 1
y = 0.6
Having found x and y, we can now calculate the area and the perimeter of the rectangle.
To find the area (A), we use the formula A = length × width. We know the expressions of length and width in terms of x and y, so we substitute these values into the expressions:
A = (4x - y) × (x + 6y)
A = (4(1.6) - 0.6) × (1.6 + 6(0.6))
A = (6.4 - 0.6) × (1.6 + 3.6)
A = 5.8 × 5.2
A = 30.16 square units
To find the perimeter (P), we use the formula P = 2 × (length + width):
P = 2 × ((4x - y) + (x + 6y))
P = 2 × ((4(1.6) - 0.6) + (1.6 + 6(0.6)))
P = 2 × (6.4 - 0.6 + 1.6 + 3.6)
P = 2 × (11.0)
P = 22.0 units
So the values of x and y are 1.6 and 0.6, respectively, the area of the rectangle is 30.16 square units, and the perimeter is 22.0 units.
