Ok, so I understand that an equation is something like 15 = 15 , and that the only criteria as far as I can tell for it being an equation is that both sides are equal to each other.
I have a few questions, the first is a question about why two equations are related to each other? For instance, if I had an equation like 15 = 15 and I subtracted another equation 7 = 7 from it, then I would have 8 = 8, which is still an equation yes, and I understand that it is possible to get back to the original equation by adding 7 to both sides again, but is it directly equal to the original equation without algebraic manipulation?
The reason that I ask this first question is that I'm curious about how after algebraic manipulation a result from a newly gotten equation directly applies to the original. If I had x + 5 = 8, and I subtracted the equation 5 = 5 from it, I would get that x = 3. I would then find that I could plug that value into the original and find that 3 + 5 = 8, which is a true equation. But why does subtracting another equation allow me to find the unknown in the original?
And finally, what makes equations special and unique from fractions in that you can add, subtract, multiply, divide, etc, all you wish from both sides of an equation but you can't do that with fractions. I understand that when you do that with a fraction, the value changes, however, when you do that with an equation doesn't the equation change? why are equations special? why can you do freely whatever you want to an equation but not a fraction?