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In all kinds of mathematical contexts substitution is used. My general question is this: what happens specifically when substituting? Why is it a valid procedure? What are the rules involved? When can or can't you substitute? What are things to be aware of when substituting?

Intuitively it seems a bit odd to me that you can just change one variable into another one.

Thanks!

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The way you pose your question, being so vague, makes it hard to pinpoint exactly things you should avoid, pitfalls you must be aware of, etc, etc, because those things are often related to the context in which you substitute.

It is common to just substitute a big expression that is repeated plenty of times with a simple variable, to make algebraic manipulation easier and more quick to perform and the we substitute back. For example

$$\left(\frac {(\cos^2(x)\cdot \sin(x) - 1)\cos(x)\sin(x)}{x^2 - 5} + 5\cos^2(x)\sin(x)\right)\cdot \cos^2(x)\sin(x)$$

To simplify that just put $a = \cos^2(x)\sin(x)$ and work from there.

Or you make a substitution that puts in evidence some expression that yields a favorable result. Again in algebraic manipulation:

$a^2 + 2a + 1 - b^2$, making $c = a + 1$ yields $c^2 - b^2 = (c - b)(c + b) = (a - b + 1)(a + b + 1)$

Some international maths olympic problems even involve finding a good substitution to make them solvable!

Substitution is also a lot used in limits, integration and derivation, in a sense because it makes it easier to manipulate smaller things as well. In those types of substitutions you must make sure you check your new bounds on the integral, for example, or that your new variable is approaching the right value.

For example a distracted student might not understand what this is

$$\lim_{x \rightarrow 0^+} (1 + x)^{1/x} $$

But making $y = 1/x $ and noticing $x \rightarrow 0^+ \iff y \rightarrow \infty $ we get

$$\lim_{y \rightarrow \infty} (1 + 1/y)^{y} = e $$

Thus substitution is really used a lot and its limits are the ones of your creativity. When you make a substitution though, just make sure everything stays coherent and that you did not lose any constraint/property.

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