Problem 60 - Algebra 1

Question

Apply the distributive property. $$(3-b) 9$$

Step-by-Step Solution

Answer

The result of applying the distributive property to \((3-b) \times 9\) is \(27 - 9b\).

Step 1: Identify the Terms Involved in the Distribution

The expression is \((3-b) \times 9\). The number outside the brackets, 9, will be distributed to each term inside the brackets, 3 and -b, individually.

Step 2: Apply the Distributive Property

For the first term inside the bracket: \(3 \times 9 = 27\). For the second term inside the bracket: \(-b \times 9 = -9b\). Combine these results for the final answer.

Step 3: Combine the Results

Combine the multiplication results together to write the final expression: \(27 - 9b\)

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. It's a way to represent quantities that can change – that is, variables – through an equation without calculating the final result. For instance, in the expression \(3-b\times 9\), both numbers and a variable are present. The number 3 is a constant, while \(b\) is a variable that can represent any number. \\ The beauty of algebraic expressions is their ability to package complex relationships between variables using simple symbols. Learning to work with these expressions is critical for solving various kinds of mathematical problems. When dealing with algebraic expressions, it's important to understand how to use operations like addition, subtraction, multiplication, and division to rearrange and simplify them.

Combining Like Terms

When simplifying algebraic expressions, one fundamental skill is combining like terms. Like terms are terms that contain the same variables raised to the same power. Only the coefficients of these terms can differ. For example, in the expression \(4a + 3a\), both terms are 'like' because they contain the same variable \(a\) to the same exponent (which is 1, even though it's not written). \\ To combine them, you would add their coefficients (the numbers in front of the variables), which in this case would result in \(7a\). This principle is central to simplifying expressions and solving equations, as it allows us to consolidate all information about a variable into a single term, making our equations much neater and easier to manage.

Multiplication Properties

Multiplication is one of the basic operations in algebra that follows several fundamental properties: \\\Commutative property: which states that changing the order of the factors does not affect the product. For instance, \(a \times b = b \times a\).\\Associative property: which allows you to regroup the factors, and it still yields the same product. For example, \(a \times (b \times c) = (a \times b) \times c\).\\Distributive property: which allows us to multiply a single term by each term within a parenthesis. For example, \(k \times (m + n) = k \times m + k \times n\).\\\The distributive property is particularly useful because it gives us a method to eliminate parentheses by distributing the multiplication over addition or subtraction within the parentheses. It also plays a critical role in operations with algebraic expressions, like expanding or factorizing them. Understanding these properties helps to perform algebraic manipulations correctly and is vital in solving equations and simplifying expressions.