Problem 18 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ -7(a+1)-9(a+4) $$

Step-by-Step Solution

Answer

The simplified expression is \\(-16a - 43\\).

Step 1: Distribute the negative sign

Start by distributing the negative sign in front of the first parenthesis, \(a+1\). This gives us the expression: \(-7a - 7\).

Step 2: Distribute the second negative number

Now distribute the \(-9\) to each term in the second parenthesis, \(a+4\). This results in \(-9a - 36\).

Step 3: Combine like terms

You now have two sets of like terms: \(-7a\) and \(-9a\), and \(-7\) and \(-36\). Combine the like terms to simplify the expression to get: \(-16a - 43\).

Key Concepts

Simplification

Simplifying algebraic expressions is all about making them easier to work with by reducing their complexity. You start with a more "complicated" expression that includes parentheses and several terms. By simplifying, you're essentially looking to reduce the expression to its most straightforward form. This is crucial because it makes further algebraic operations, like solving equations, much simpler.

Consider the original expression \( -7(a+1) - 9(a+4) \).This expression involves several operations including multiplication and addition. The goal is to simplify by solving these operations step-by-step. Start by removing the parentheses through distribution and then combine like terms. It ultimately results in a much simpler expression: \( -16a - 43 \).

Simplification is an essential skill in algebra that helps make longer and more complex expressions manageable and sets the stage for more advanced topics in mathematics.

Distributive Property

The Distributive Property is a fundamental algebraic property used to simplify expressions that involve parentheses. It is crucial when you need to eliminate parentheses and distribute a factor across terms inside parentheses. This property states that for any numbers or expressions, \( a(b+c) = ab + ac \).Hence, a number or variable outside the parentheses is multiplied by each term inside the parentheses.

In the expression \( -7(a+1) - 9(a+4) \), we apply the distributive property to get rid of the parentheses:

• First, distribute \( -7 \) to each term inside \( (a+1) \). This results in \( -7a - 7 \).
• Then, distribute \( -9 \) across \( (a+4) \), leading to \( -9a - 36 \).

The distributive property breaks down more extensive multiplications into smaller, more manageable pieces, helping further simplify expressions. It acts as a powerful tool for unraveling complicated terms tied up in parentheses, enabling smoother and more straightforward calculations.

Combining Like Terms

Combining like terms is the process of merging terms that have the same variable component and the same degree. It's a vital step in algebra to streamline expressions after applying operations like the distributive property.

In our expression \( -7a - 7 - 9a - 36 \), we identify the like terms:

• The \( a \) terms are \( -7a \) and \( -9a \). By adding these coefficients, we end up with \( -16a \).
• For the constant terms, we have \( -7 \) and \( -36 \). Combine these to get \( -43 \).

Thus, the expression becomes \( -16a - 43 \).Combining like terms reduces the clutter in our expression, ensuring it's as clear and concise as possible. This process is indispensable because it generalizes numbers and variables into a more usable format, facilitating easier problem-solving in subsequent algebraic operations.