SOLUTION: Question: The sum of the reciprocal of an integer and the reciprocal of six-more-than-that-integer is the same as 5 / 36. I have no idea how to start to set this up or solve it.

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Question 205516: Question: The sum of the reciprocal of an integer and the reciprocal of six-more-than-that-integer is the same as 5 / 36.
I have no idea how to start to set this up or solve it. I am so confused.
Thank you for the help
Answer by Targetweek(62) About Me  (Show Source):
You can put this solution on YOUR website!
so %281%2Fx%29%2B%281%2Fy%29+=+5%2F36
and y+=+x%2B6
now you have a systems of equations and can solve it
first substitute the second equation into the first
1.%281%2Fx%29%2B%281%2F%28x%2B6%29%29+=+5%2F36
2. Get the same denominator
a. %281%2Fx%29%2B%28x%2Fx%29%281%2F%28x%2B6%29%29+=5%2F36
-> %281%2Fx%29%2B%28x%2Fx%28x%2B6%29%29+=+5%2F36
b. %28%28x%2B6%29%2F%28x%2B6%29%29%281%2Fx%29%2B%28x%2Fx%28x%2B6%29%29+=+5%2F36
-> %28x%2B6%29%2Fx%28x%2B6%29%2Bx%2Fx%28x%2B6%29+=+5%2F36
3. Simplify by addiing together
-> %282x%2B6%29%2Fx%28x%2B6%29+=+5%2F36
4. Simplify further
a. %28%28x%28x%2B6%29%29%2F1%29%28%282x%2B6%29%2Fx%28x%2B6%29%29+=+%28%28x%28x%2B6%29%29%2F1%29%285%2F36%29
-> 2x%2B6+=+5x%28x%2B6%29%2F36
b. %2836%2F1%29%282x%2B6%29+=+%2836%2F1%29%285x%28x%2B6%29%2F36%29
-> %282x%2B6%29%2F36+=+5x%28x%2B6%29
c. 2x%2F36%2B6%2F36+=+5x%2Ax+%2B+5x%2A6
-> x%2F18%2B3%2F18+=+5x%5E2%2B30x
-> %28x%2B3%29%2F18+=+5x%5E2%2B30x
d. %2818%2F1%29%28%28x%2B3%29%2F18%29+=+%2818%2F1%29%285x%5E2%2B30x%29
-> x%2B3+=+90x%5E2%2B540x
5. Now get everything over to one side and simplify
a. 90x%5E2%2B539x-3+=+0
6.Now use the quadratic formula to solve for x
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 90x%5E2%2B539x%2B-3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28539%29%5E2-4%2A90%2A-3=291601.

Discriminant d=291601 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-539%2B-sqrt%28+291601+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28539%29%2Bsqrt%28+291601+%29%29%2F2%5C90+=+0.0055606995840675
x%5B2%5D+=+%28-%28539%29-sqrt%28+291601+%29%29%2F2%5C90+=+-5.99444958847296

Quadratic expression 90x%5E2%2B539x%2B-3 can be factored:
90x%5E2%2B539x%2B-3+=+90%28x-0.0055606995840675%29%2A%28x--5.99444958847296%29
Again, the answer is: 0.0055606995840675, -5.99444958847296. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+90%2Ax%5E2%2B539%2Ax%2B-3+%29


and there are your two answers
... actually it's not... those aren't integers... are you sure you input the problem verbatim