SOLUTION: I need help solving this equation, we have been working on factoring and I can't seem to find a solution for this equation, please help! Thanks! Solve: 6/(y-2)+7/(y-8)=(y-1)/

Question 553439: I need help solving this equation, we have been working on factoring and I can't seem to find a solution for this equation, please help! Thanks!

Solve: 6/(y-2)+7/(y-8)=(y-1)/(y-8)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
your original equation is:
6/(y-2) + 7/(y-8) = (y-1)/(y-8)
multiply both sides of this equation by (y-2)(y-8) to get:
6(y-8) + 7(y-2) = (y-1)(y-2)
simplify to get:
6y - 48 + 7y - 14 = y^2 - 3y + 2
simplify by combining like terms to get:
13y - 62 = y^2 - 3y + 2
subtract 13y and add 62 to both sides of the equation to get:
0 = y^2 - 16y + 64
commute to get:
y^2 - 16y + 64 = 0
that's your quadratic equation which can now be factored to:
(y-8)^2 = 0 which becomes:
(y-8)(y-8) = 0 which yields:
y = 8
that's your answer until you plug that answer back into the original equation.
then you get a nasty surprise because x = 8 is undefined in the original equation because it makes the denominator equal to 0.
that means that there is no real solution to your original equation.
x = 8 is a good answer for y^2 - 16y + 64 because it does yield the root at x = 8.
when you graph the quadratic equation, you get something that looks like this:

note that i had to replace y with x in order to graph it.
note that the graph of the equation crosses the x-axis at x = 8.
that's the root of the quadratic equation.
-----
when you graph the original equation, you get something very different that looks like this:

note that the graph does cross the x-axis at x = 8.
the problem is that the value of y is undefined in the original equation when x = 8 resulting in what is called a hole.
a hole is a gap in the value of y.
you have values of y all around that spot, but you do not have a value for y when x = 8.
note again that your original equation was expressed with the variable of y but that i changed y to x in order to be able to graph it.
when you graph the equation, x is the independent variable and y is the dependent variable.
that has something to do with the software only being able to recognize x on the horizontal axis and y on the vertical axis.
if you were drawing the draph manually, then you could have assigned horizontal to z and vertical to z or any other letter.
you do have an answer for your quadratic equation.
that answer is 8.
you do not have an answer for the original equation because x = 8 (rather y = 8) is not defined in the original equation.

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