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Understanding like terms in algebra is crucial for simplifying expressions. Like terms are terms that have identical variable parts, meaning they have the same variables raised to the same power. For example, in the terms \(3x^2\) and \(5x^2\), both have the variable part \(x^2\), thus they are like terms. However, \(3x^2\) and \(5x\) are not like terms because the variables are raised to different powers.

In the given exercise, we see \(9t\) and \(12t\) as like terms because the variable \(t\) is the same in both terms and not raised to any power. This aspect is what makes them 'like' and eligible to be combined through mathematical operations such as addition or subtraction.

Simplifying algebraic expressions is a vital process in algebra. To simplify an algebraic expression, you combine like terms, which involves adding or subtracting the coefficients while keeping the variable part of the term untouched. Simplification makes the expression easier to read and work with.

To illustrate, let’s simplify \(2a + 3a\). Since \(2a\) and \(3a\) are like terms (they both have \(a\) as the variable part), we add the coefficients \(2\) and \(3\) to get \(5a\). Now, the expression is in its simplest form. Simplification can involve combining multiple like terms and taking care to preserve the correct signs (positive or negative) for each term. Always keep in mind to match only the like terms, since mismatched terms cannot be simplified in this manner.

Subtracting coefficients is part of the process of combining like terms in algebra. Coefficients are the numerical part of algebraic terms that are attached to variables. When you have like terms and you need to subtract them, you only subtract the coefficients and keep the variable part unchanged.

For instance, consider the terms \(9t\) and \(12t\). To subtract \(12t\) from \(9t\), we subtract the coefficients \(12\) from \(9\), yielding \(9 - 12 = -3\). After computing the coefficients, we combine the result with the variable to get \( -3t\). It is essential to pay attention to the signs of the coefficients as they determine the outcome of the subtraction. Notably, in subtracting coefficients, if both terms have negative coefficients, you are essentially adding the absolute values of the numbers, and vice versa.