Problem 46 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$4(5 x-3+7 y)$$

Step-by-Step Solution

Answer

The algebraic expression \(4(5x - 3 + 7y)\) without parentheses, using the distributive property, is \(20x - 12 + 28y\).

Step 1: Identify the terms inside the parentheses

In the given algebraic expression \(4(5x - 3 + 7y)\), \(5x\), \(-3\), and \(7y\) are the terms inside the parentheses.

Step 2: Apply the distributive property

Applying distributive property means you multiply the term outside the parentheses (in this case, 4) to each term inside the parentheses separately, and then add (or subtract, depending on the term) the results. So, \(4 \cdot 5x\), \(4 \cdot -3\), and \(4 \cdot 7y\) are obtained.

Step 3: Simplify the expression

Doing the individual multiplications, you get \(20x\), \(-12\), and \(28y\). The expression without parentheses is then \(20x - 12 + 28y\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. These are used to represent a quantity or a relationship abstractly. For example, the expression \(5x - 3 + 7y\) is an algebraic expression where:

• \(5x\) means 5 times some variable \(x\).
• \(-3\) is a constant term that doesn't change.
• \(7y\) indicates 7 times some variable \(y\).

Variables like \(x\) and \(y\) stand in for numbers we do not know yet. They allow us to solve problems and equations that can be applied in various real-life contexts. Thus, understanding algebraic expressions is crucial for learning broader mathematical concepts.

Simplification

Simplification is the process of making an algebraic expression more manageable or easier to understand. It involves reducing the expression into its simplest form by executing operations and combining like terms when possible.Consider the expression given: \(4(5x - 3 + 7y)\). Using the distributive property, we multiply each term inside the parentheses by 4:

• First, calculate \(4 \cdot 5x = 20x\).
• Next, \(4 \cdot (-3) = -12\).
• Then, \(4 \cdot 7y = 28y\).

After performing these multiplications, we rewrite the expression without parentheses as \(20x - 12 + 28y\). This process of simplification helps to more easily visualize and work with the expression for further operations such as solving, graphing, or substituting values for the variables.

Terms in Algebra

In algebra, terms are the building blocks of expressions. A term can be a single number, a variable, or the product of numbers and variables. Understanding terms is vital because it determines how you handle operations such as addition, subtraction, and multiplication within an expression.For the expression \(4(5x - 3 + 7y)\):

• \(5x\) is a term consisting of the coefficient 5 and the variable \(x\).
• \(-3\) is a constant term with no variable attached.
• \(7y\) includes the coefficient 7 and the variable \(y\).

Each term is separated by addition or subtraction signs. After distributing the 4, these terms become \(20x, -12,\) and \(28y\) respectively. Recognizing the distinct terms in an expression is essential for applying algebraic operations correctly and simplifying expressions effectively.