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**epson1** I have also wondered another thing: If we have an equation that includes an unknown (possibly several unknowns), then how can we say that adding or multiplying both sides of the equation with the same number preserves the equality sign. For example, take the first equation: 3x+1=2x-3. Now we don't know yet which value of x satisfies the equation. If we now substract for example the number "1" from both sides of the equation, we would get 3x=2x-4. Notice that we still have the "=" sign between those two expressions, altough we don't even know if they are equal, because we don't know the correct number x. Or do we assume here that "x" is the correct solution to the equation, and then this presumption would allow us to preserve the "=" sign when substracting 1 from both sides? Tough if we did this presumption, then we would have a contradiction if the equation didn't have any solution at all.