Problem 37 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 9(4 y-3) $$

Step-by-Step Solution

Answer

Question: Multiply the expression 9(4y - 3) and simplify the result. Answer: The simplified expression is 36y - 27.

Step 1: Distributive Property

Multiply each term inside the parentheses by 9: $$ 9(4y - 3) = 9 \cdot 4y - 9 \cdot 3 $$

Step 2: Perform Multiplications

Next, perform the multiplications: $$ 36y - 27 $$

Step 3: Combine Like Terms

Since there are no like terms in the expression, the final answer is: $$ 36y - 27 $$

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. In simple terms, it's a mix of these elements that represents a particular value or set of values. For example, in the expression \(9(4y - 3)\), there are numbers (9 and 3), a variable \(y\), and operations of multiplication and subtraction.

• Variables are symbols that represent unknown values, commonly letters like \(x, y, z\).
• Coefficients are numbers multiplied with the variables, such as 4 in \(4y\).
• Constants are numbers without variables attached, such as 3.

Understanding algebraic expressions is crucial in working with equations and formulas, as these expressions help in modeling real-world scenarios.

Like Terms

Like terms are terms in algebraic expressions that have identical variable parts, even if their coefficients are different. They are essential for simplifying expressions as they can be added or subtracted from each other.
For instance, in \(36y - 27\), the terms "36y" and "27" are not like terms. They do not have the same variable part, so we can't combine them.

• Terms like \(5x\) and \(3x\) are like terms because both have the variable \(x\).
• Without the same variable, terms like \(4x\) and \(3y\) are not alike.

Recognizing and combining like terms simplify expressions and make equations easier to handle.

Multiplication in Algebra

Multiplication in algebra involves multiplying numbers, variables, and expressions, utilizing properties such as the distributive property. This property is key in expanding expressions, allowing us to remove parentheses by distributing a factor to each term within. For instance, using the distributive property, \(9(4y - 3)\) expands to \(9 \cdot 4y - 9 \cdot 3\).
This step-by-step expansion can be broken down as:

• Multiply the coefficient outside the parentheses by each term inside: \(9 \cdot 4y = 36y\)
• Multiply again for the next term: \(9 \cdot -3 = -27\)
• Combine the results: \(36y - 27\)

Mastering multiplication in algebra helps in solving equations, expanding expressions, and working through complex algebraic problems.