SOLUTION: Why is 2/5x - 2/3y = 5 not a linear equation?

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Question 345200: Why is 2/5x - 2/3y = 5 not a linear equation?
Found 2 solutions by checkley77, jsmallt9:
Answer by checkley77(12844) About Me  (Show Source):
You can put this solution on YOUR website!
2/5x - 2/3y = 5
-2y/3=-2x/5+5 multiply by -3/2
y=(-2x/5)(-3/2+5(-3/2)
y=6x/10-15/2
y=3x/5-7.5
+graph%28+300%2C+200%2C+-6%2C+15%2C+-10%2C+10%2C+3x%2F5+-7.5%29+ (graph 300x200 pixels, x from -6 to 15, y from -10 to 10, 3x/5 -7.5).
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"A linear equation is the form ax+by=c, with a,b & c real numbers & a & b not both zero."

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Please put your denominators in parentheses. Without them we cannot tell if the equation is
%282%2F5%29x+-+%282%2F3%29y+=+5
or
2%2F%285x%29+-+2%2F%283y%29+=+5
These are very different equations. The first one is a line. And another tutor has provided a solution based on the first interpretation of your equation.

The second equation is not a line. And since your question implies that the equation is not supposed to be a line, I assume that the second interpretation is correct. If this is so, then you should have typed:
2/(5x) - 2/(3y) = 5

I will now provide an explanation for why the second equation is not a line. Start by multiplying both sides by the Lowest Common Denominator (LCD) of the two denominators. (Usually LCD's are used to add and subtract fractions. But, as you will soon see, they are useful in other ways too.) The LCD is 15xy:
15xy%282%2F%285x%29+-+2%2F%283y%29%29+=+16xy%285%29
On the left side we need to use the Distributive Property to multiply:
15xy%282%2F%285x%29%29+-+15xy%282%2F%283y%29%29+=+80xy
30xy%2F%285x%29+-+30xy%2F%283y%29+=+80xy
Both fractions reduce. (This is the usefulness of multiplying by the LCD.)
6y+-10x+=+80xy

Linear equations (equations of lines) can be written in the form Ax + By = C or y = mx+b. They do NOT have any xy terms! (Nor do they have x%5E2, y%5E2, etc. They just have plain x terms and/or plain y terms.) Because of the 80xy, this is not a linear equation!