ABCD is a parallelogram with angle B= 120, angle C= 60, angle D= 2x+5y, and angle A= 4x+y. Find the values of x and y. Check your solution, only an algebraic solution will be accepted.
Please solve this
To find the values of x and y in the parallelogram ABCD, we need to use the properties of a parallelogram.
In a parallelogram, opposite angles are equal. So, angle D equals angle B, and angle A equals angle C.
We can start by equating angle D and angle B:
2x + 5y = 120 ...(1)
Next, we equate angle A and angle C:
4x + y = 60 ...(2)
Now we have a system of two equations. We can solve this system of equations using algebraic techniques.
Let's solve Equations (1) and (2) using the method of substitution:
From Equation (2), we can isolate y:
y = 60 - 4x
Substitute this value of y into Equation (1):
2x + 5(60 - 4x) = 120
Simplify the equation:
2x + 300 - 20x = 120
Combine like terms:
-18x + 300 = 120
Move the constant term to the other side:
-18x = 120 - 300
-18x = -180
Divide both sides by -18:
x = -180 / -18
x = 10
Now substitute the value of x back into Equation (2) to find y:
4(10) + y = 60
Simplify:
40 + y = 60
Subtract 40 from both sides:
y = 60 - 40
y = 20
Therefore, x = 10 and y = 20.
To check the solution, substitute the values of x and y back into the original equations and verify that both sides are equal:
For Equation (1):
2x + 5y = 120
2(10) + 5(20) = 120
20 + 100 = 120
120 = 120
For Equation (2):
4x + y = 60
4(10) + (20) = 60
40 + 20 = 60
60 = 60
Both sides of the equations are equal, so the solution x = 10 and y = 20 is valid.