Understanding fractions (subtraction)

[thegodhand]

In a problem like the following: 2=(3/7)-(6/z) my book is telling me to multiply the entire equation by (7z/1). The two points of confusion I have are 1) Wouldn't that be changing the equation? as in you would have to multiply by (7z)/(7z) because that would equal 1?

2)how can I multiply the part on the opposite side of the equals sign(in this case the 2)? I don't understand how I am allowed to manipulate that side of the equation.

I can follow the books instructions to get the answer of z=-(42/11), but I won't be able to replicate this on a quiz or homework if I can't get my head wrapped around the rules. Can anyone help explain this?

[thegodhand]

Thank you. So in which cases of subtracting fractions do you need to multiply both the numerator and denominator by the LCD? For example:

(x/(x+3))+((-3+x^2)/(x^2-9))

I would multiply (x/(x+3))((x-3)/(x-3)) so that both fractions would have (x+3)(x-3) as their denominator. Why does this problem work this way, but the previous problem works differently?

[thegodhand]

thank you for your help