Problem 18 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) 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 through the Parentheses

First, distribute the '-7' across the terms inside the first set of parentheses '(a+1)'. You multiply '-7' by both 'a' and '1'.\[-7(a+1) = -7a - 7\]Next, distribute '-9' across the terms inside the second set of parentheses '(a+4)'. Multiply '-9' by 'a' and '4'.\[-9(a+4) = -9a - 36\]

Step 2: Combine the Distributed Terms

Now, combine the two expressions that resulted from the distribution step. This gives:\[-7a - 7 - 9a - 36\]

Step 3: Combine Like Terms

Identify and combine the like terms from the expression from Step 2. The like terms are the terms involving 'a' and the constant terms.Combine '-7a' and '-9a':\[-7a - 9a = -16a\]Combine '-7' and '-36':\[-7 - 36 = -43\]Now, rewrite the expression with the combined terms:\[-16a - 43\]

Key Concepts

Distributing Terms

When simplifying algebraic expressions, one of the primary steps is distributing terms. This step involves taking a number or variable outside a set of parentheses and multiplying it by each term within the parentheses. This process ensures every part of the expression is accounted for and helps eliminate the parentheses.

Let's take the expression \(-7(a+1)-9(a+4)\). Here, we start by distributing \(-7\) to each term inside the first parentheses \((a+1)\).

• First, multiply \(-7\) by \(a\), resulting in \(-7a\).
• Then, multiply \(-7\) by \(1\), giving \(-7\).

Thus, the expression \(-7(a+1)\) simplifies to \(-7a - 7\).

Next, apply the same concept to the second set of parentheses \((a+4)\) using \(-9\).

• Multiply \(-9\) by \(a\) to get \(-9a\).
• Multiply \(-9\) by \(4\) for a result of \(-36\).

Thus, \(-9(a+4)\) becomes \(-9a - 36\). Combining distributed results, our expression is \(-7a - 7 - 9a - 36\). The parentheses are now removed, paving the way for further simplification.

Combining Like Terms

Once distribution is complete, you'll often find multiple similar terms that can be combined in an algebraic expression. This process, known as combining like terms, streamlines the expression and makes it easier to manage.

In our example \(-7a - 7 - 9a - 36\), we have to identify and group the like terms:

• Terms involving the variable \(a\): these are \(-7a\) and \(-9a\). Combining them gives \(-16a\), calculated by adding the coefficients \(-7\) and \(-9\).
• Constant numbers: \(-7\) and \(-36\). Adding these yields \(-43\).

By combining like terms, the expression significantly reduces complexity. After performing these calculations, you simplify the expression to \(-16a - 43\).

This simplification condenses all terms and helps achieve a cleaner, more easily interpreted algebraic expression. Combining like terms is a crucial step in algebra simplification.

Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest form. By distributing terms and combining like terms, we've streamlined our expression from \(-7(a+1) - 9(a+4)\) to \(-16a - 43\).

The goals of algebraic simplification are varied:

• Eliminate any unnecessary or redundant expressions.
• Make the expression easier to understand and solve in later parts or related problems.
• Prepare the ground for more advanced operations, such as factorization or solving equations.

Upon final simplification, ensure there are no more like terms. Also, confirm that any distributed or additional operations are complete.

In this way, algebraic simplification ensures that you are left with the most efficient and tidy form of the expression, making it ready for any subsequent mathematical operations.