Problem 69 - Algebra 1

Question

Apply the distributive property. $$7(3-2 y)$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(21 - 14y\).

Step 1: Apply the distributive property

First, distribute the number 7 over each term inside the brackets. We do this by multiplying each term in the brackets with 7 separately. This gives us: \( 7 \cdot 3 - 7 \cdot 2y \)

Step 2: Simplify the expressions

Now, multiply the terms to simplify the expressions. We get: \(21 - 14y\)

Key Concepts

Simplifying Algebraic Expressions

Understanding how to simplify algebraic expressions is a foundational skill in algebra. It involves combining like terms and reducing expressions to their simplest form. This process often requires the use of the distributive property, where you multiply a single term by each of the terms within a set of parentheses.

For instance, in the given exercise where we have to simplify the expression \(7(3 - 2y)\), the first step is to apply the distributive property. This means we take the number outside the parentheses, which is 7 in this case, and multiply it by each term inside the parentheses separately. By doing this, each part of the expression gets its turn to be multiplied by 7, leading to \(7 \times 3 - 7 \times 2y\).

After applying the distributive property, we simplify by performing the multiplication, which results in \(21 - 14y\). This final expression cannot be simplified further because there are no like terms to combine, so we’ve decomposed the exercise into its simplest form, making the concept easier to grasp.

Multiplying Variables

When multiplying variables, it's essential to remember that variables are placeholders for numbers, and the rules of multiplication apply to them just as they do to numbers. In algebra, when multiplying a variable by a number, you multiply the coefficient (the number in front of the variable) by the given number.

For example, in the expression \(7 \times -2y\), the variable \(y\) has a coefficient of -2. To multiply this by 7, we simply multiply the coefficients together, ignoring the variable for a moment. The multiplication results in \(-2 \times 7\), which equals \(-14\). We then bring back the variable, leading to the term \(-14y\).

This is a key step in simplifying algebraic expressions, and mastering it will help students to tackle more complex problems with confidence. Variables can seem intimidating, but once you understand the basics of multiplying them, they become much less so.

Applying Algebraic Properties

Algebraic properties are rules that apply to numbers and variables. They are critical tools in simplifying expressions and solving equations. The distributive property, which was applied in our exercise, is one of these fundamental properties.

Understanding the Distributive Property

The distributive property states that multiplying a number by a sum (or a difference) is the same as multiplying the number by each term in the sum (or difference) and then adding (or subtracting) the products. As seen in the exercise, we distributed the number 7 across the terms within the parentheses, resulting in \(7 \times 3 - 7 \times -2y = 21 - 14y\).

Significance of Algebraic Properties

Algebraic properties are the backbone of simplifying expressions. They provide a structured approach to manipulating equations and ensuring that every step taken is logically sound. By systematically applying these properties, students can navigate through algebraic problems efficiently, ensuring a deep understanding and the ability for application in various mathematical scenarios.