Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$11(6 a+3 b)+4(12 a+5 b)$$

Step-by-Step Solution

Answer

The simplified algebraic expression is \(114a + 53b\)

Step 1: Apply Distributive Property

Multiplication is distributive over addition, so each number outside of the parentheses should be multiplied by each term inside the parentheses. This results in: \(66a + 33b + 48a + 20b\)

Step 2: Combine Like Terms

After successfully applying the distributive property from Step 1, the next step is to group and add like terms (terms with the same variable). This results in: \(66a + 48a + 33b + 20b\)

Step 3: Sum the Coefficients

The final step is to sum the grouped coefficients. From previous step, we had grouped coefficients of 'a' together and 'b' together. Hence the expression simplifies to: \(114a + 53b\)

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that simplifies expressions and solves equations. This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. This is expressed as: \[ a(b + c) = ab + ac \].

To see this in action, consider the expression given in the exercise: \[ 11(6a + 3b) + 4(12a + 5b) \].
Each term outside the parentheses is multiplied by every term within. For \( 11(6a + 3b) \), this means you will multiply \( 11 \) by \( 6a \) and then by \( 3b \), resulting in \( 66a + 33b \).

Similarly, for \( 4(12a + 5b) \), multiply \( 4 \) by \( 12a \) and then by \( 5b \), leading to \( 48a + 20b \).

The beauty of the distributive property is in its ability to break down complex expressions into easier-to-handle parts, streamlining calculations and paving the way for further simplification.

Combining Like Terms

Combining like terms is a process used to simplify algebraic expressions, making them easy to work with by merging terms that have the same variable(s). Like terms have identical variables raised to the same powers, but can have different coefficients—for example, \( 5x \) and \( 7x \) are like terms, while \( 5x \) and \( 5y \) are not.

In our exercise, the result after applying the distributive property was: \( 66a + 33b + 48a + 20b \).
We group the like terms. Terms with 'a' are \( 66a \) and \( 48a \). Terms with 'b' are \( 33b \) and \( 20b \).

By combining, or adding, these like terms, we get:

• \( 66a + 48a = 114a \)
• \( 33b + 20b = 53b \)

Combining like terms helps reduce the complexity of the expression, turning it into a neat and more manageable form for further calculations or applications.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. A variable is a symbol, often a letter, that represents an unknown or variable number.

The expression provided in the exercise, \( 11(6a + 3b) + 4(12a + 5b) \), is an algebraic expression consisting of several components:

• The coefficients: 11 and 4, multiply the terms within the parentheses.
• The terms inside the parentheses: \( 6a + 3b \) and \( 12a + 5b \) are also algebraic expressions.
• The operations: addition and multiplication, dictate the relationships between the numbers and variables."

When simplifying algebraic expressions, the goal is to condense the expression as much as possible without changing its value. Applying the distributive property and combining like terms, as shown in the exercise, are vital techniques in achieving simplification, ultimately resulting in a simpler expression: \( 114a + 53b \).

Understanding the structure and simplification of algebraic expressions is essential for solving equations and making connections between mathematical ideas.