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Substitution is an essential technique in algebra where we replace a variable with its given value. Imagine variables as placeholders or empty boxes that can be filled with specific numbers. When solving an algebraic fraction like \(\frac{d}{12}\) and you're told that \(d = 60\), we simply replace the \(d\) with 60, making our expression \(\frac{60}{12}\).

Why do we do this? Substitution gives us a concrete number that we can work with, paving the way for further operations such as division or simplification. It’s very much like following a recipe: if you're told to use 'a cup of sugar', that's exactly what you do - only in this case, 'a cup of sugar' is our given value, such as 60.

Division operations are fundamental in simplifying algebraic fractions. In the step-by-step solution, performing the division \(\frac{60}{12}\) is like dividing a pizza into 12 equal slices and then seeing how many whole pizzas are made if you have 60 slices.

Division tells us how many times one number is contained within another. In our example, we're asking, 'how many times does 12 fit into 60?' The answer helps simplify the fraction to its smallest form, resulting in a whole number or a simpler fraction. Through division, we understand the relationship between numbers and reduce expressions to their most straightforward form.

Simplifying expressions is the process of reducing an algebraic expression to its simplest form. When we tackled the problem \(\frac{60}{12}\), after substituting the value of \(d\) and performing the division, we simplified the fraction. But what does simplifying really mean?

In general, to simplify an expression, we carry out all possible divisions and cancellations until we cannot go further without changing the value of the expression. A simplified expression is easier to understand and work with. It's like cleaning up your room so you can see everything clearly. After dividing 60 by 12, we were left with 5, a much neater and simpler expression than where we started. Whether we're dealing with fractions or more complex algebraic equations, simplifying is about making the expression as clear and concise as possible.