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When we simplify algebraic expressions, one of the most fundamental steps is combining like terms. Like terms are terms within an expression that have the same variable raised to the same power. For example, in the given exercise, we are looking at the terms 9a and -4a. These are considered like terms because they both contain the variable a raised to the same power, which is 1 (though typically we don't write the exponent when it is 1).

To combine these terms, we focus on the algebraic coefficients, which are the numerical parts of the terms (in this case, 9 and -4). Essentially, we handle the coefficients as we would with usual addition or subtraction of numbers, while keeping the variable part unchanged. The result is a single term that represents the sum or difference. Simply put, to combine 9a and -4a, we calculate 9 - 4 to get an answer of 5a.

This simplification is like gathering all of your apples into one basket – you count how many you have, regardless of where they came from, because they are all apples (like terms).

The algebraic coefficients are crucial when manipulating algebraic expressions. They are numerical factors that multiply a variable or variables. In the example of 9a and -4a, the coefficients are 9 and -4, respectively. It's important to pay attention to the sign accompanying each coefficient, as it determines how the terms will be combined — whether they will be added or subtracted.

When combining like terms, it’s just the coefficients that are added or subtracted; the variables remain the same. As seen in the exercise, we subtract the coefficient -4 from 9, which simplifies to 5a. Coefficients tell us 'how many' of the variables we have, and they are pivotal in stating the quantity of the term. This allows for terms to be combined in a meaningful way.

Subtracting polynomials involves distributing negative signs and combining like terms. Although the current example doesn’t involve a polynomial, the principle is similar — the subtraction of polynomials involves looking for like terms across different parts of a polynomial and applying the rules of subtraction. Like terms, which contain the same variables and exponents, are identified and subtracted by their coefficients.

For example, if you had to subtract 3x^2 + 2x - 5 from 6x^2 - x + 4, you'd rearrange the polynomials to align like terms and then subtract coefficient by coefficient: (6 - 3)x^2 + (-1 - 2)x + (4 - (-5)), simplifying to 3x^2 - 3x + 9.

This process is similar to removing a certain number of objects from groups of the same type. The structured approach of aligning terms and then subtracting ensures that each term is correctly handled according to its type (degree and variable).