If ON=2x-3, LM=7x+2, NM=x+6, and OL=6y-3, find the values of x and y for which LMNO must be parallelogram.

To be a parallogram, ON = LM

2x-3 = 7x+2
-5x = 5
x = -1

and 6y-3 = x+6
6y = -1+6+3
y = 8/6 = 4/3

To determine the values of x and y for which LMNO must be a parallelogram, we need to set up some conditions.

Condition 1: Opposite sides of a parallelogram must be equal in length.
Therefore, we have the following equations:
ON = LM (i)
NM = OL (ii)

Using the given expressions for ON, LM, NM, and OL, we can substitute in the values:

2x - 3 = 7x + 2 (from equation i)
x + 6 = 6y - 3 (from equation ii)

Let's solve these two equations simultaneously to find the values of x and y:

2x - 3 = 7x + 2
Rearranging the equation:
2x - 7x = 2 + 3
-5x = 5
Dividing both sides by -5:
x = -1

Substituting this value of x in equation ii:
-1 + 6 = 6y - 3
5 = 6y - 3
6y = 5 + 3
6y = 8
Dividing both sides by 6:
y = 8/6
y = 4/3

Therefore, x = -1 and y = 4/3 are the values for which LMNO must be a parallelogram.

To determine the values of x and y for which LMNO must be a parallelogram, we need to compare the opposite sides of the quadrilateral and check if they are equal.

First, let's compare the lengths of the opposite sides LM and NO:
LM = 7x + 2
NO = ON + NM = 2x - 3 + (x + 6) = 3x + 3

For LMNO to be a parallelogram, LM must be equal to NO. So we have:
7x + 2 = 3x + 3

Now, let's solve this equation for x:
7x - 3x = 3 - 2
4x = 1
x = 1/4

Now that we have found the value of x, we can substitute it back into any of the given equations to find the value of y. Let's use the equation OL = 6y - 3:
OL = 6y - 3
2x - 3 = 6y - 3
2(1/4) - 3 = 6y - 3
1/2 - 3 = 6y - 3
-2.5 = 6y - 3
2.5 = 6y
y = 2.5/6
y = 5/12

The values of x and y for which LMNO must be a parallelogram are x = 1/4 and y = 5/12.