Problem 54 - Prealgebra

Question

Use the distributive property to combine similar terms. \(-7 a-5 a\)

Step-by-Step Solution

Answer

The expression simplifies to \(-12a\).

Step 1: Identify Like Terms

In the expression \(-7a - 5a\), notice that both terms have the variable \(a\). This makes them like terms, meaning they can be combined.

Step 2: Apply the Distributive Property

Use the distributive property, which states \(a(b + c) = ab + ac\), in reverse to factor the expression. Since the expression \(-7a - 5a\) is already subtracted, combine the coefficients first: \((-7) + (-5) = -12\).

Step 3: Combine Coefficients and Factor

Combine the coefficients from Step 2 into a single term: \(-12a\). The expression can now be simplified to \(-12a\).

Key Concepts

Understanding Similar Terms

Similar terms, also known as like terms, are terms in an algebraic expression that have the same variable raised to the same power. Unlike terms might have different coefficients, but they must involve the same variable and power to be combined. In the exercise given,

• The terms 7a and 5a are similar because they both have the variable 'a'.
• Such similarities allow us to combine them easily.

When identifying similar terms, always check for matching variables and exponents, regardless of their coefficients.

Being comfortable with recognizing similar terms is crucial for simplifying expressions more effectively as it helps in collecting similar components together.

Factoring in Expressions

Factoring, in the context of algebra, is the process of breaking down expressions into more manageable pieces or finding common factors that can be used to simplify the expression.

• Think of factoring as looking for common elements that can 'group' terms together.
• In the expression 7a - 5a, we are essentially factoring by leveraging the distributive property in reverse.

By applying the distributive property, which states that \(a(b + c) = ab + ac\), we rewrite the expression by factoring out the common variable 'a' from both terms,

• resulting in \(a(-7 + (-5))\).
• Through this, we simplify the expression into clearer parts, making it easier to solve or further simplify.

Factoring simplifies expressions and is an integral step whenever you're prepping terms for combination or simplification.

How to Combine Like Terms

Combining like terms involves consolidating similar terms in an expression by adding or subtracting their coefficients. This process reduces expressions to simpler forms. Consider the expression

• 7a - 5a, both terms have the same variable 'a', enabling us to combine them by operating on their coefficients:
• Negative seven plus negative five \((-7) + (-5)\).

Calculating these coefficients yields \(-12\), thus simplifying the expression to \(-12a\). This consolidation helps in solving equations or simplifying expressions and is particularly useful in various algebraic manipulations.

By mastering the technique of combining like terms, you reduce clutter in mathematical expressions and find solutions faster and more accurately.