Problem 43 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ -3(a-6) $$

Step-by-Step Solution

Answer

Answer: The simplified expression is $-3a + 18$.

Step 1: Distribute the constant term to every element in the parentheses

Multiply -3 by each term within the parentheses: $$ -3(a-6) = (-3 \cdot a) + (-3 \cdot -6) $$

Step 2: Perform the multiplication

Now, perform the multiplication: $$ (-3 \cdot a) + (-3 \cdot -6) = -3a + 18 $$

Step 3: Combine like terms

In this expression, there are no like terms to combine: $$ -3a + 18 $$ So the final simplified expression is: $$ -3a + 18 $$

Key Concepts

Multiplication

Multiplication is one of the basic operations in algebra and involves combining quantities. When you multiply a number by another, you are essentially adding it to itself a specified number of times. In the context of our exercise, the multiplication process helps us simplify expressions. Here, we see

• The number -3 is multiplied with each term inside the parentheses, and
• Each term is treated separately.

The multiplication results in \(-3 \times a\) producing \(-3a\), and \(-3 \times (-6)\) producing \(+18\), following the rules of positive and negative sign interactions. Always ensure you perform each multiplication separately and accurately.

Distribution

The distributive property is a handy algebraic tool for simplifying expressions involving multiplication over addition or subtraction. This property states that:

• \(a(b + c) = ab + ac\)

In simple terms, you distribute, or "spread out," the multiplication across items in parentheses. In our exercise:

• The term \(-3(a-6)\) uses distribution to become \((-3 \cdot a) + (-3 \cdot -6)\).
• Each term inside is multiplied separately by -3.

This method is crucial when dealing with polynomials and can make complex expressions much easier to handle.

Like Terms

The concept of like terms helps simplify algebraic expressions by combining terms with identical variable parts. In our exercise, once we distribute and multiply, we look for like terms, which are terms with the same variable raised to the same power. For example:

• Terms like \(-3a\) and \(+18\) are not like terms because one contains the variable \(a\), and the other is a constant.

After multiplication and distribution, there are no like terms to combine in the expression \(-3a + 18\). Always identify like terms to simplify expressions further, reducing them to a cleaner form.