91Ó°ÊÓ

When simplifying algebraic expressions, one of the first steps is to combine like terms. Like terms are terms that have the exact same variables raised to the same power. In other words, they're the building blocks that share a common variable part. For instance, in the expression \( -12y + 5y \), both terms are 'like' because they're both multiples of \( y \).

To combine them, simply add or subtract their numerical coefficients—the numbers in front of the variables. Think of it much like grouping similar items together. If you have 12 apples and someone gives you 5 more, you have to sum up the total number of apples. Likewise, combining \( -12 \)(apples) with \( 5 \)(apples) gives you \( -7 \)(apples), or \( -7y \) in algebraic terms.

In algebra, coefficients are the numbers that multiply the variables. They provide the 'how much' for each variable in an equation or expression. Take the example \( -12y + 5y \). Here, \( -12 \) and \( 5 \) are the coefficients of the variable \( y \).

Understanding coefficients is crucial because they determine the value and behavior of algebraic expressions. If the coefficient is positive, it indicates an addition; if negative, a subtraction. When simplifying expressions, your job is to manipulate these coefficients to find a simplified, combined term. Remember that only coefficients can be added or subtracted from each other; variables remain unchanged unless they are like terms.

Simplifying an algebraic expression breaks down into a series of logical and sequential steps. It can be likened to following a recipe—each step must be done in order to achieve the desired outcome. Here are the usual simplification steps distilled into manageable parts:

• Identify like terms: Scan the expression for terms that have the same variable parts.
• Combine like terms: Add or subtract the coefficients of these terms.
• Simplify coefficients: Once like terms are combined, multiple terms can reduce into a single term with a new coefficient.
• Re-evaluate: After each simplification, check if there are more like terms to combine or any other operations to perform.
• Final expression: Once all like terms are combined and simplified, you end up with the most reduced form of the original expression.

Following these steps diligently will yield a correct and simplified algebraic expression, making understanding and solving algebra problems much more digestible.