Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ (7-m) 4 $$

Short answer

The expression \( (7-m) 4 \), rewritten without parentheses using the distributive property, is \(28 - 4m\).

Step-by-step solution

Distribution of the Multiplication

To begin with, start by distributing the multiplication of 4, to each term inside the parentheses: \(4 \cdot 7\) and \(4 \cdot -m\).

Perform the Multiplications

Perform each multiplication: \(4 \times 7 = 28\) and \(4 \times -m = -4m\).

Write Down the Final Expression

With the multiplication performed, the final expression should be \(28 - 4m\).

Key concepts

Rewriting Expressions

Mastering the art of rewriting expressions is essential in algebra, as it allows students to simplify complex equations and solve problems efficiently. The process involves manipulating an expression into an equivalent form, making it easier to work with. For instance, if faced with an expression wrapped in parentheses, like the one in the exercise \( (7-m) 4 \), applying the distributive property permits us to rewrite it without parentheses. This step is crucial as it prepares the groundwork for further operations such as combining like terms or solving for variable 'm'.

Understanding how to properly distribute a number over terms within parentheses is a foundational skill. Using the steps outlined in the solution, which detail multiplying each term inside the parentheses by the number outside, we can transmute \( (7-m) 4 \) into a linear expression. This simplified form, \(28 - 4m\), reveals more clearly the relationship between the constants and variables, essential for grasping the underlying algebraic structure.

Algebraic Expressions

Algebraic expressions are the backbone of algebra and serve as a universal language to represent real-world problems in mathematical terms. An algebraic expression can include numbers, variables, and operations. In the exercise, \( (7-m) 4 \), '7' and '4' are constants, 'm' is a variable, and the operations involved are multiplication and subtraction.

Grasping the syntax of algebraic expressions is akin to learning a new language; once familiar, it enables students to tackle a range of mathematical challenges. When we rewrite expressions—transforming \( (7-m) 4 \) into \(28 - 4m\)—we are, in fact, translating a phrase from a compact form to an expanded version that often provides more insight. This expanded form is particularly advantageous when solving equations or evaluating expressions for given values of variables, showcasing the expression's practical utility in a variety of contexts.

Properties of Operations

The properties of operations are the rules that dictate how numbers and variables behave when we manipulate them in algebra. These rules are the distributive property, the commutative property, the associative property, and the properties of identity and inverse operations. The distributive property, which is used in the exercise, is especially important as it allows for the multiplication of a number by a sum or difference within parentheses.

The correct application of the distributive property is critical when rewriting expressions. For example, when we distribute the '4' across the terms '7' and '-m' in the expression \( (7-m) 4 \), we are applying the distributive property: \(4 \times 7 + 4 \times (-m)\) which simplifies to \(28 - 4m\). Understanding this property enables students to confidently expand, factor, and simplify algebraic expressions. It's like having a powerful tool in their mathematical toolkit, prepared to dismantle complex operations into more manageable parts, paving the way for solution finding and problem-solving in algebra.