Problem 42 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ -6(y+7) $$

Step-by-Step Solution

Answer

Based on the given step by step solution, the expression -6(y+7) simplifies to -6y - 42 when distributed and combined.

Step 1: Distribute the -6

Multiply the -6 by each term inside the parentheses: $$ -6(y+7) = -6\cdot y + (-6) \cdot 7 $$

Step 2: Simplify the multiplication

Calculate the multiplication from the previous step: $$ -6\cdot y + (-6) \cdot 7 = -6y -42 $$

Step 3: Combine like terms

There are no further like terms to combine. The final expression is: $$ -6y - 42 $$

Key Concepts

Combining Like Terms

When working with algebraic expressions, you'll often need to simplify them to make them easier to understand or to prepare them for solving equations. One key method of simplification is combining like terms. Like terms are terms that have the same variable raised to the same power. Only the coefficients (the numbers in front of the variables) can be different.

Let's consider the expression obtained after distributing in our example: \( -6y - 42 \). Here, the term \( -6y \) has a variable \(-y\), while \( -42 \) is a constant term without a variable. Since there's only one term with the variable \(-y\), there are no other like terms to combine with \( -6y \). Thus, the expression is already as simplified as it can be in terms of combining like terms.

However, if we had an expression like \( -6y + 3y - 42 \), we could combine the like terms \( -6y \) and \( 3y \) to get \( -3y - 42 \), simplifying the expression further. Remember, the ability to identify and combine like terms is crucial for simplifying expressions efficiently.

Simplifying Expressions

Simplifying expressions is a cornerstone of algebra which makes equations easier to work with. Simplification can involve multiple steps, including distributing multiplication over addition or subtraction, combining like terms, and reducing expressions to their simplest form.

For instance, recall our example \( -6(y+7) \). The expression was simplified by distributing the multiplication of \( -6 \) across the terms within the parentheses. This distribution gave us \( -6y - 42 \) as the simplified form. At this point, the expression cannot be simplified further, as noted in the example because there are no like terms to combine.

When simplifying, always look out for opportunities to combine like terms or further reduce the expression. It's a process of peeling away the layers until the expression is as streamlined as possible. In practice, simplification may involve factoring, canceling common factors, or reducing fractions when applicable.

Multiplication in Algebra

Multiplication in algebra works similarly to multiplication in basic arithmetic, but it can involve variables as well as numbers. A fundamental rule in algebraic multiplication is the distributive property, which allows you to multiply a single term by a group of terms within a set of parentheses.

The distributive property states that \( a(b + c) = ab + ac \). Even when dealing with negative numbers or subtraction, the property applies—for example, \( a(b - c) = ab - ac \). In our textbook exercise, we used the distributive property to expand \( -6(y+7) \) which resulted in \( -6y -42 \).

This property is especially useful for simplifying expressions before further algebraic manipulation. It's important to ensure each term inside the parentheses is multiplied by the term outside — a common mistake is to neglect multiplying every term, leading to incorrect simplifications.