Question 306433: How do you find the values of x and y that ensure eace quadrilateral is a parallelogram? The top is 2x-4, the bottom is 10, the left side is -3y and the right side is 12-y.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! If each quadrilateral is a parallelogram, then the opposite sides have to be parallel to each other.
The top is 2x-4
The bottom is 10
The left side4 is -3y
The right side is 12-y.
?????
The bottom has to be parallel to the top.
I have no idea what you mean by the bottom is 10, but let's just say that the bottom is 2x + 10.
This give it the same slope as the top.
The left side is -3y and the right side is 12 - y.
?????
-3y means y = -1/3
12 - y means y = 12.
That's possible since the slope of the equations would both = 0.
To graph this equation, we would get graph the equations:
y = 2x-4
y = 2x - 10
y = -1/3
y = 12
The graph would look like this:
You can see that we have a parallelogram because the top and bottom sides are parallel to each other, and the left and right sides are also parallel to each other.
I'm not sure if this is the parallelogram you were thinking about, but this is a prallelogram.
The slopes of the opposite sides have to be equal to each other.
The y intercepts can be anything as long as they are different from each other.
The equation of a straight line is y = mx + b.
m is the slope.
b is the y-intercept which is the value of y when x = 0.
The equations we just showed on the graph are:
y = 2x-4
y = 2x - 10
y = -1/3
y = 12
Another set of equations where the slopes are the same but the y-intercepts are different would be:
y = 2x-4
y = 2x+15
y = -5
y = 5
A graph of these equations would look like this:
The parallelogram is positioned in a different spot, but it is still a parallelogram and the parallel sides are still oriented in the same direction.
This is because the y-intercepts have changed, but the slopes have remained the same.
The parallelograms we are talking about are the spaces that are bounded by the parallel lines.
The corner point of the parallelograms would be the intersections of the 2 sets of parallel lines.
Answer by ikleyn(53266)  (Show Source):
You can put this solution on YOUR website! .
How do you find the values of x and y that ensure  that quadrilateral is a parallelogram?
The top is 2x-4, the bottom is 10, the left side is -3y and the right side is 12-y.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Regarding this problem, tutor @Theo expresses perplexity.
Meanwhile, its meaning is clear and the solution is simple.
The necessary and sufficient condition for a quadrilateral to be a parallelogram is that
the opposite sides are congruent, i.e. have equal length.
It gives us two equations
2x-4 = 10 for the top and the bottom sides,
-3y = 12 - y or the left and right sides.
From the first equation
2x = 10 + 4 = 14 ---> x = 14/2 = 7.
From the second equation
-3y + y = 12 ---> -2y = 12 ---> y = 12/(-2) = -6.
ANSWER. x = 7, y = -6.
Solved.
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