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Imagine you have a few apples, say 3, and you give away more than you have, let's say 9. How many apples do you have now? This is the basic idea behind coefficients subtraction in algebra. It works just like subtracting regular numbers, but it includes a variable, like 'x' in our current example. When you see an expression such as \(3x - 9x\), you're essentially looking at a situation where you start with 3 of something (3x) and then you're taking away 9 of the same thing (9x).

The process is straightforward: you take the number in front of the variable (the coefficient) for the first term (3 in this case) and subtract the coefficient of the second term (9 in this case) from it. For our example, it's simply a matter of calculating \(3 - 9\), which yields \(-6\). So, \(3x - 9x\) simplifies to \(-6x\), meaning if you had 3 'x's and took away 9 'x's, you would end up with negative 6 'x's. It's important to remember that when subtracting coefficients, variables must be the same; what if we had 3 oranges and tried to take away 9 apples? It gets confusing! Hence, like terms are vital in coefficients subtraction.

Algebraic expression simplification is like tidying up a room – you're organizing and combining items to make it look neat and manageable. In algebra, simplifying means to combine like terms and perform any arithmetic operations to make the expression as easy as possible to understand or work with.

Using our exercise \(3x - 9x\) as an example, simplification involves identifying the like terms which are both terms containing the variable 'x'. Since they're like terms, we can perform arithmetic operations - in this case, subtraction. The result, as we determined earlier, is \(-6x\). That's our simplified expression. The goal is always to boil down the expression to its most basic form without changing its value. Simplification may include various operations like addition, subtraction, multiplication, division, or factoring, depending on the complexity of the expression.

When faced with an algebraic expression, 'combining like terms' is akin to pairing socks from a pile of laundry – you group the matching pairs together. In algebra, terms are 'like' if they have the same variables raised to the same power. Numbers without variables, also known as constant terms, pair with other constant terms.

Let's look at the terms in our expression, \(3x - 9x\). Both terms have the same variable 'x' raised to the first power (which is typically unwritten). They’re like terms! We can combine them by adding or subtracting their coefficients. Since our operation is subtraction, we subtract 9 from 3, getting \(-6\), and keep the variable 'x' with it, thus simplifying to \(-6x\).

Why is combining like terms important?

It allows us to simplify expressions to their easiest form making subsequent calculations like solving equations or evaluating expressions far more manageable. Always look out for like terms; they are your first step towards simplification in algebraic expressions.