The sides of a square are given are in cm the sides are 4y-3, 4x+1, 5y-3x, y+3x. Find xand y and the area of the square
Since the sides of the square are given as 4y-3, 4x+1, 5y-3x, and y+3x, we can set them equal to each other.
4y-3 = 4x+1
5y-3x = y+3x
Now we can solve these equations to find the values of x and y.
From the first equation:
4y-3 - 4x = 1
4y - 4x = 4
y - x = 1
From the second equation:
5y-3x - y = 3x
4y - 4x = 3x
y - x = 3/4
Now we have a system of equations to solve:
y - x = 1
y - x = 3/4
Since these equations do not have a unique solution, it means that there is no unique value for x and y. This means that the given side lengths do not form a square.
Therefore, we cannot find the area of the square.
To find the values of x and y, we need to equate the expressions for opposite sides of the square.
Given the sides:
Side 1 = 4y - 3
Side 2 = 4x + 1
Side 3 = 5y - 3x
Side 4 = y + 3x
Since opposite sides of a square are equal, we can set up equations:
Side 1 = Side 3:
4y - 3 = 5y - 3x (equation 1)
Side 2 = Side 4:
4x + 1 = y + 3x (equation 2)
Now, let's solve the equations to find the values of x and y.
From equation 1:
4y - 5y = -3x + 3
-y = -3x + 3
y = 3x - 3 (equation 3)
From equation 2:
4x - 3x = y - 1
x = y - 1 (equation 4)
Now, substitute equation 3 into equation 4 to eliminate y:
x = (3x - 3) - 1
x = 3x - 4
Solving this equation gives us:
-2x = -4
x = -4 / -2
x = 2
Using this value of x, we can substitute it into equation 4 to find y:
y = 2 - 1
y = 1
So, x = 2 and y = 1.
To find the area of the square, we can use any side length since they are all equal. Let's use side 1 = 4y - 3:
Side length = 4(1) - 3
= 4 - 3
= 1 cm
Area of the square = (side length)^2
= (1)^2
= 1 cm^2
Therefore, the values of x and y are x = 2 and y = 1, and the area of the square is 1 square cm.
To find the values of x and y and the area of the square, we need to solve the given equations.
Given:
Side 1: 4y - 3
Side 2: 4x + 1
Side 3: 5y - 3x
Side 4: y + 3x
Since a square has all sides equal, we can set up the following equations:
Equation 1: 4y - 3 = 4x + 1
Equation 2: 4x + 1 = 5y - 3x
Equation 3: 5y - 3x = y + 3x
Now, let's solve these equations step by step:
Step 1: Solve Equation 1 for x in terms of y:
4y - 3 = 4x + 1
4x = 4y - 4
x = (4y - 4)/4
Simplifying further, we get:
x = y - 1
Step 2: Substitute the value of x in Equation 2:
4x + 1 = 5y - 3x
4(y - 1) + 1 = 5y - 3(y - 1)
Distributing and simplifying, we get:
4y - 4 + 1 = 5y - 3y + 3
4y - 3 = 2y + 3
2y = 6
y = 3
Step 3: Substitute the value of y in Equation 3:
5y - 3x = y + 3x
5(3) - 3x = 3 + 3x
15 - 3x = 3 + 3x
6x = 12
x = 2
So, we have found that x = 2 and y = 3.
To find the area of the square, we can use any of the side lengths. Let's use Side 1: 4y - 3.
Area = (Side)²
= (4y - 3)²
= (4 * 3 - 3)² [Substitute the values of x = 2 and y = 3]
= (12 - 3)²
= 9²
= 81 square units
Therefore, the value of x is 2, the value of y is 3, and the area of the square is 81 square units.