Question - Solving for Rectangle Dimensions and Properties
Solution:
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 + 12. The shorter sides (width) of the rectangle are equal, so we have: x + 6y = 2x + 2Now we'll solve these two equations simultaneously.From equation 1:4x - y = 3x + 1x - y = 1 (Subtract 3x from both sides)From equation 2:x + 6y = 2x + 26y = 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 + 26(x - 1) = x + 26x - 6 = x + 26x - x = 2 + 65x = 8x = 8 / 5x = 1.6Now, we substitute the value of x back into the equation y = x - 1 to find y:y = 1.6 - 1y = 0.6Having 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 × 5.2A = 30.16 square unitsTo 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 unitsSo 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.
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