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When we work with algebraic expressions, identifying like terms is a crucial first step in simplification. Like terms are terms that have the exact same variable components raised to the same powers. An easy way to spot them is by looking at the letters and their exponents, ignoring any coefficients or numbers attached.

For example, in the expression 12y and -15y, both terms have the variable y and do not have any exponents, which makes them like terms. They can be combined through addition or subtraction. It is important to remember that only like terms can be combined this way; terms with different variables or exponents should be treated as distinct and not combined.

Once like terms have been identified in an algebraic expression, we can proceed to subtract the coefficients if the operation indicated is subtraction. The coefficient is the numerical part that is multiplied by the variable in an algebraic term. In our expression 12y - 15y, the coefficients are 12 and 15, respectively.

To subtract the coefficients, focus only on these numbers and perform the normal subtraction operation: \(12 - 15 = -3\). Then, the variable part will be simply carried over to your result. Remember that subtraction of coefficients should preserve the sign of each term; a common mistake is to ignore the negative sign, which can lead to incorrect results. Always consider the order of terms, as subtraction is not commutative, meaning that \(12 - 15\) is not the same as \(15 - 12\).

An arithmetic expression is a combination of numbers, operations signs, and sometimes variables, arranged with the conventions of arithmetic operations like addition, subtraction, multiplication, and division. In contrast to algebraic expressions, arithmetic expressions do not include variables, but when they do, they become algebraic.

In the context of algebra, when we simplify an arithmetic expression containing variables, we aim to reduce it to its simplest form by performing operations like adding or subtracting coefficients of like terms. Keep in mind that to truly simplify an arithmetic expression involving variables, you first need to combine like terms, if any, then perform arithmetic operations to obtain the most reduced form possible.