Problem 75 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$-9 a+11 a d-35 a+a d$$

Step-by-Step Solution

Answer

The simplified expression is \(-44a + 12ad\).

Step 1: Identify Like Terms

First, identify terms with the same variables and the same exponents. In this expression, the terms are \(-9a\), \(11ad\), \(-35a\), and \(ad\). The like terms are \(-9a\) and \(-35a\), both of which contain the variable \(a\) with no other variables attached, and \(11ad\) and \(ad\), both containing the variables \(a\) and \(d\).

Step 2: Combine Like Terms with 'a' Only

Combine the coefficients of the terms with only \(a\). So, you have \(-9a - 35a\). Add the coefficients: \(-9 + (-35) = -44\). So, this becomes \(-44a\).

Step 3: Combine Like Terms with 'ad'

Combine the coefficients of the terms with both \(a\) and \(d\). The expression is \(11ad + ad\). Remember that \(ad\) is equivalent to \(1ad\). So, add the coefficients: \(11 + 1 = 12\). Thus, this becomes \(12ad\).

Step 4: Write the Simplified Expression

The simplified expression is a combination of the results from Steps 2 and 3. Therefore, the simplified expression is \(-44a + 12ad\).

Key Concepts

Identifying Like Terms

In algebraic expressions, identifying like terms is the first crucial step towards simplification. Like terms are terms that have identical variables raised to identical powers. It's important to develop an eye for spotting these components, as they pave the way to effectively combining them.
For instance, in the expression \(-9a + 11ad - 35a + ad\), we notice that:

• \(-9a\) and \(-35a\) share the variable \(a\), each raised to the power of one, with no other variables attached.
• Similarly, \(11ad\) and \(ad\) feature the combination of \(a\) and \(d\) raised to the first power.

Identifying like terms allows you to streamline the expression by collating coefficients (the numbers in front of the variables) for terms that are alike. This organized approach sets you up for straightforward simplification later on.

Simplification of Algebraic Expressions

Simplification involves reducing the expression to its simplest form by combining the identified like terms. Once you've pinpointed the like terms, focus on their coefficients. The coefficients are the numerical parts that can be added or subtracted based on the like terms they accompany.
Consider the expression:

• Start with combining \(-9a\) and \(-35a\): here, you add their coefficients: \(-9 + (-35) = -44\).
• Next, address \(11ad\) and \(ad\). Recall \(ad\) implies \(1ad\); adding the coefficients yields \(11 + 1 = 12\).

The key to successful simplification is ensuring variables and powers match before altering coefficients. Ultimately, this leads to a cleaner, more manageable expression, which for our example is \(-44a + 12ad\). Simplification showcases the beauty of algebra: transforming complexity into clarity.

Algebraic Coefficients

Algebraic coefficients are the numerical components attached to variables in an expression, and they play a pivotal role during simplification. Understanding how to handle these coefficients is vital in combining like terms effectively.
Every algebraic term comprises a coefficient and a variable part. Let's break down the expression \(-9a + 11ad - 35a + ad\):

• \(-9\) and \(-35\) are coefficients of \(a\), wherein \(-9a\) and \(-35a\) are like terms.
• Similarly, \(11\) and the implicit \(1\) are coefficients of \(ad\), appearing in the terms \(11ad\) and \(ad\) respectively.

To simplify, we compute the sums of these coefficients, keeping the variable parts unchanged:

• The sum of \(-9\) and \(-35\) becomes \(-44a\).
• For the terms \(11ad\) and \(1ad\), add their coefficients to get \(12ad\).

Through mastering coefficients, you can efficiently adjust and streamline algebraic expressions, leading you directly to their simplest form.