Find the values of x and y that make the quadrilateral a parallelogram. Show all your work. a parallelogram is drawn with opposite sides of 4 times y minus 1 and 2 times y plus 7. The second set is 3 times x plus 1 and 4 times x minus 2
To determine the values of x and y that make the quadrilateral a parallelogram, we need to set the opposite sides of the quadrilateral equal to each other.
Given that the opposite sides are:
Side 1: 4y - 1
Side 2: 2y + 7
Side 3: 3x + 1
Side 4: 4x - 2
Now, let's set Side 1 equal to Side 3, and Side 2 equal to Side 4:
4y - 1 = 3x + 1 (Equation 1)
2y + 7 = 4x - 2 (Equation 2)
Next, we can solve the system of equations to find the values of x and y.
First, solve Equation 1 for x:
3x = 4y - 2
x = (4y - 2) / 3
Then, substitute this expression for x into Equation 2:
2y + 7 = 4((4y - 2) / 3) - 2
Simplify Equation 2 further:
2y + 7 = (16y - 8) / 3 - 2, let's get rid of the fractions by multiplying both sides of the equation by 3:
6y + 21 = 16y - 8 - 6
Combine like terms:
6y + 21 = 16y - 14
Move the variables to one side and the constants to the other:
6y - 16y = -14 - 21
Combine like terms again:
-10y = -35
Divide both sides by -10:
y = (-35) / (-10)
y = 3.5
Now, substitute the value of y back into Equation 1 to find x:
4(3.5) - 1 = 3x + 1
Simplify the equation:
14 - 1 = 3x + 1
Combine like terms:
13 = 3x + 1
Solve for x:
3x = 13 - 1
3x = 12
x = 12 / 3
x = 4
Therefore, the values of x and y that make the quadrilateral a parallelogram are x = 4 and y = 3.5.
To determine the values of x and y that make the quadrilateral a parallelogram, we need to equate the opposite sides of the quadrilateral.
Given:
Opposite sides:
Side 1: 4y - 1
Side 2: 2y + 7
Side 1: 3x + 1
Side 2: 4x - 2
For a parallelogram, opposite sides must be equal.
Equating the opposite sides, we have:
4y - 1 = 2y + 7 [equation 1]
3x + 1 = 4x - 2 [equation 2]
Solving equation 1:
4y - 1 = 2y + 7
4y - 2y = 7 + 1
2y = 8
y = 8/2
y = 4
So, we have found the value of y, which is 4.
Now, substitute the value of y into equation 2:
3x + 1 = 4x - 2
Simplifying the equation:
3x - 4x = -2 - 1
-x = -3
x = -3/-1
x = 3
Therefore, the values of x and y that make the quadrilateral a parallelogram are x = 3 and y = 4.