Problem 67 - Algebra 2

Question

Simplify. $$ \frac{1}{3}(15 a+9 b)-\frac{1}{7}(28 b-84 a) $$

Step-by-Step Solution

Answer

The simplified expression is \( 17a - b \).

Step 1: Distribute the First Term

Distribute the factor of \( \frac{1}{3} \) to each term inside the parentheses. Apply the distributive property: \( \frac{1}{3}(15a) + \frac{1}{3}(9b) \). This simplifies to \( 5a + 3b \).

Step 2: Distribute the Second Term

Distribute the factor of \( \frac{1}{7} \) to each term inside the parentheses, remembering the minus sign: \( -\frac{1}{7}(28b) + \frac{1}{7}(84a) \). This simplifies to \( -4b + 12a \).

Step 3: Combine Like Terms

Now, combine the results from Steps 1 and 2. Our expression is now: \( 5a + 3b + 12a - 4b \).

Step 4: Simplify the Expression

Combine like terms to simplify the expression: \( 5a + 12a = 17a \) and \( 3b - 4b = -b \). The simplified expression is \( 17a - b \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to break down and simplify expressions. It is used to multiply a single term by terms inside a set of parentheses. This property states that for any numbers or variables,

• \( a(b + c) = ab + ac \)

In simpler terms, you multiply the term outside the parentheses by each term within the parentheses separately and then add or subtract the results. This technique is handy for expressions that are not easily simplified all at once.

In the original exercise, the distributive property was used twice. First, with the factor \( \frac{1}{3} \) applied to \( 15a + 9b \), resulting in \( 5a + 3b \). Then, the factor \( -\frac{1}{7} \) was applied to \( 28b - 84a \), resulting in \( -4b + 12a \). By distributing and simplifying early, you can make the entire process of solving algebraic problems much easier.

Combining Like Terms

Combining like terms is another essential step in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. Only the coefficients of these terms are different, and these are the only parts of the terms that you add or subtract.

• Examples of like terms: \( 2x \) and \( -5x \)
• Non-like terms: \( 3x \) and \( 3y \)

To combine like terms, we add or subtract their coefficients. This helps in reducing the expression to its simplest form, enabling easier interpretation and manipulation for further calculations.

In the original problem, the like terms were \( 5a + 12a \) and \( 3b - 4b \). By combining the \( a \) terms, we get \( 17a \), and by combining the \( b \) terms, we get \( -b \). Properly combining like terms is crucial for reducing expressions to their simplest form.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations. Unlike an equation, an algebraic expression does not have an equality sign. It merely represents a value and is used extensively in various mathematical calculations and problem-solving.

• Example: \( 3x + 4 \)
• Components: Variables like \( x \), coefficients like \( 3 \), and constants like \( 4 \).

Algebraic expressions are used to model real-world situations, as they can represent unknown values and help solve problems involving these unknowns through manipulation and simplification.

In the exercise given, the expression \( \frac{1}{3}(15a+9b)-\frac{1}{7}(28b-84a) \) is an algebraic expression. Simplifying such expressions allows you to find equivalent forms that are easier to interpret. Mastering the principles of working with algebraic expressions is foundational in algebra, enabling more advanced mathematical reasoning and problem-solving.