So what are the postulates/axioms that allow us to do the basic arithmetic operations to an equation, and then we come up with a new, but still equivalent equation with the first one? In practise: Why does an equation, which has been procecced using the same operations on both sides of the equation, have the same, yet only the same solutions as the original equation? And I want to tell you that I already know the "scale demonstration"/"balanced steelyard", which isn't the answer I'm lookin for. I need something more profound explanations.
How is it exactly proven using the field axioms? I'm a high school student, so my understanding will be limited to what i have learned so far about maths( mostly one or two variable calculus, geometry, trig, algebra etc). Anyway I would be interested to see how the proofs go, and what kind of axioms are being used in them. This issue is also something that has brought me so much headache during my high school years.
here. But you should be careful about saying that "an equation, which has been processed using the same operations on both sides of the equation, [has] the same, yet only the same solutions as the original equation". That is not always true. For example, the equation has only one solution. But if you apply the operation of squaring to both sides of the equation then you get , in which a new solution has been introduced.
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.
Watch Video on Substitution Property of Equality - Math Help on Yahoo! Video.
To see how that principle justifies the procedures used in solving equations, take the above example 3x+1=2x–3. Informally, we want to subtract 2x from both sides to conclude that x+1=–3. To justify that operation, first look at the left side of the equation. We know from rules of algebra (associativity, commutativity) that 3x+1–2x = x+1. The substitution property says that if 3x+1=2x–3 then we can substitute 2x-3 for 3x+1 in any equation, and the result will be the same. It follows that if (3x+1)–2x = x+1 then (2x-3)–2x = x+1. But 2x–3–2x = –3. Conclusion: x+1=–3.
The next operation (subtracting 1 from each side of that last equation) is justified in a similar way. We know that (x+1) – 1 = x. If x+1=–3 then the principle of substitution says that we can substitute –3 for x+1 and deduce that (–3) – 1 = x, in other words x=–4.
How can you tell 3x+1=2x-3, before you have made sure x can be only -4? I mean you can't state that 3x+1=2x-3, unless you know that it really is true. For example we could say x+1=x, which is a false statement in the field of real numbers.
And thanks for introducing me the substitution property, because it is clearly the one of the axioms used in equations.
That still doesn't prove that there is a solution. Plugging into the original equation proves that is the only solution.
As for the equation , assume there is a solution. Then we would have 1=0, which is false, so there are no solutions.