Problem 43 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(5(7-4 y)\)

Step-by-Step Solution

Answer

The simplified form of the expression is \(35 - 20y\).

Step 1: Apply the Distributive Property

The distributive property states that a term outside the parentheses can be multiplied individually with each term inside the parentheses. In this exercise, distribute the 5 to both terms inside the parentheses: 7 and \(-4y\). This gives:\[5 \times 7 + 5 \times (-4y)\]

Step 2: Perform the Multiplication

Now, perform the multiplication for each term from Step 1:\[5 \times 7 = 35\]\[5 \times (-4y) = -20y\]

Step 3: Write the Expression

Combine the results of Step 2 into a single expression:\[35 - 20y\]

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, and division) that represent a value. Think of them as math sentences that can include unknown elements, known as variables. In our exercise, we encountered an algebraic expression inside the parentheses: \(7 - 4y\). Here, 7 is a constant (a number), and \(y\) is a variable that stands in for an unknown number.

Expressions like these are the building blocks of algebra. They allow us to model real-world situations and solve problems where some quantities are unknown. Understanding the components, such as terms and coefficients, is essential. For example, in \(-4y\), \(-4\) is the coefficient telling us how many times \(y\) is added or subtracted. Recognizing these parts helps in manipulating and solving algebraic expressions effectively.

Simplifying Expressions

Simplifying expressions means transforming them into a simpler form without changing their value. It's like cleaning up the expression to make it easier to understand or solve. In the context of our problem, simplifying involved using the distributive property to eliminate the parentheses and combine like terms.

The expression \(5(7-4y)\) was simplified by distributing the 5 to both inside terms, transforming it into \(35 - 20y\). Simplifying often involves a sequence of steps:

• Removing parentheses by using the distributive property.
• Combining like terms, which are terms with the same variable raised to the same power.
• Reducing to the simplest form, which may look cleaner and give insight into the behavior of the expression.

Mastering these steps gives you a powerful tool for tackling a variety of algebra problems, leading to solutions more swiftly.

Multiplication in Algebra

In algebra, multiplication follows specific rules that remain consistent with basic arithmetic but involve variables. When multiplying algebraic expressions, you need to adhere to these rules to avoid errors. In the exercise, the multiplication of \(5\) over each term within the parentheses demonstrated this process. Here's how it works:

When you multiply a constant by a term inside parentheses, like \(5 \times 7\), you simply multiply the numbers, yielding 35. For terms with variables, such as \(5 \times (-4y)\), multiply the numbers first (5 and -4), then append the variable \(y\), resulting in \(-20y\).

Remember, multiplication in algebra can involve:

• Multiplying coefficients (numbers before variables) directly.
• Maintaining the variable as part of the product.
• Respecting the rules of negative and positive numbers.

This understanding ensures accurate manipulations of expressions, allowing you to simplify or solve them effectively. It’s an essential skill for advancing in algebra and tackling more complex mathematical challenges.