Problem 34 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$(y+2) 10$$

Step-by-Step Solution

Answer

The equivalent expression is \(10y + 20\).

Step 1: Understand the problem

We have the expression \((y + 2) 10\). Our goal is to use the Distributive Property to rewrite this as an equivalent expression. The Distributive Property states that \(a(b + c) = ab + ac\).

Step 2: Identify the parts of the expression

In the expression \((y + 2) 10\), \(a = 10\), \(b = y\), and \(c = 2\). We can apply the Distributive Property to this expression.

Step 3: Apply the Distributive Property

Distribute the \(10\) over the terms inside the parentheses: \(10(y + 2) = 10 \times y + 10 \times 2\).

Step 4: Simplify the products

Calculate each product from the previous step: \(10 \times y = 10y\) and \(10 \times 2 = 20\).

Step 5: Write the equivalent expression

Combine the terms to write the equivalent algebraic expression: \(10y + 20\).

Key Concepts

Equivalent Expressions

In algebra, creating equivalent expressions is an essential skill. It involves transforming one expression into another while preserving its value and meaning. For instance, when you use the Distributive Property, you can rewrite an expression like

• (y + 2) 10

in a different form, but with equal value. By distributing the 10, you transform it into

• 10y + 20.

Although these expressions look different, they are equivalent. This means they represent the same mathematical quantity. Equivalent expressions are very useful because they allow us to simplify or manipulate expressions into more workable forms.
Another common scenario is finding equivalent expressions by combining like terms or factoring. Every time you rewrite one algebraic expression while maintaining its value, you're finding an equivalent expression.

Algebraic Expression

An algebraic expression is a mathematical phrase that can involve numbers, variables, and operation symbols. In the given example,

• (y + 2) 10 and the resulting 10y + 20

are both algebraic expressions. The key elements of an algebraic expression include:

• **Variables**, like y - they represent unknown values or quantities that can change.
• **Constants**, such as 2 or 10 - these are fixed numbers with a definite value.
• **Operators**, like + or \( imes \) - these indicate the mathematical operations to perform.

Understanding the components of an algebraic expression helps us to manipulate or solve them. Different algebraic expressions might look completely different at a glance, but they can often be simplified or rewritten to reveal their equivalent expressions.

Simplifying Expressions

Simplifying expressions is the art of making them as straightforward as possible. In the given expression,

• 10(y + 2)

we use the Distributive Property to simplify it into

• 10y + 20.

Simplifying an expression involves:

• Distributing any coefficients, like the 10.
• Combining like terms. This means adding up any terms that have the same variable part.
• Performing arithmetic operations on numbers.

Simplification makes expressions easier to work with, often leading to quicker calculations or clearer understanding of the ideas they represent. In many cases, simplified expressions are also easier to solve as they reduce the complexity of the problem.