Problem 52 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$10(r-s)$$

Step-by-Step Solution

Answer

The expression is equivalent to \(10r - 10s\).

Step 1: Understanding the Distributive Property

The Distributive Property in algebra allows you to multiply a single term by each term inside a parenthesis. It can be written as \(a(b + c) = ab + ac\), where \(a\) is distributed to both \(b\) and \(c\).

Step 2: Applying the Distributive Property

We have the expression \(10(r - s)\). Apply the Distributive Property by multiplying 10 with each term inside the parenthesis: \(r\) and \(-s\).

Step 3: Distributing 10 to Each Term

First, distribute 10 to \(r\): \(10 \cdot r = 10r\).Next, distribute 10 to \(-s\): \(10 \cdot (-s) = -10s\).

Step 4: Combining the Products

Combine the products obtained from distributing 10 to each term. This gives us the expression \(10r - 10s\).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols such as addition, subtraction, multiplication, and division. For example, an expression like \(10(r-s)\) contains:

• Numbers: These are the constants that multiply the variables. In our example, the number is 10.
• Variables: These are symbols used to represent unknown values, such as \(r\) and \(s\).
• Operations: These connect the numbers and variables. Here, we have subtraction inside the parentheses.

Algebraic expressions can be simple or complex, depending on the number of terms and operations involved. They are foundational in algebra as they allow for representing real-world problems and performing calculations. Understanding the structure of these expressions helps in manipulating them using various mathematical properties, like the Distributive Property.

Multiplication in Algebra

Multiplication in algebra works similarly to arithmetic multiplication, but with the inclusion of variables. When multiplying algebraic expressions, each term in one expression is multiplied by each term in another. This can make expressions more complex. The distributive property is one of the key tools used here, especially when dealing with expressions in parenthesis.
The distributive property helps us see that \(a(b+c)\) means multiplying both \(b\) and \(c\) by \(a\). It's helpful when you want to "distribute" the multiplication across terms within parentheses.
Let's apply this to our earlier example:

• Given \(10(r-s)\), you distribute 10 to both \(r\) and \(-s\) separately.
• This becomes \(10 \cdot r\) and \(10 \cdot (-s)\).
• When multiplying numbers, think about signs. Positive \(\times\) Positive = Positive, but Positive \(\times\) Negative = Negative.
• Thus, that results in \(10r - 10s\).

Using multiplication to distribute terms helps break down operations and makes it possible to simplify complex expressions.

Simplifying Expressions

Simplifying an expression means rewriting it in its simplest form. The goal is to make it as easy to work with as possible. Simplification often involves combining like terms and applying arithmetic operations while adhering to mathematical rules and properties.
When you start with an expression like \(10(r-s)\), using the distributive property is the first step to simplify.

• First, you multiply through each of the terms: \(10 \cdot r = 10r\) and \(10 \cdot (-s) = -10s\).
• The result is the new, equivalent expression: \(10r - 10s\).

This expression, \(10r - 10s\), cannot be simplified further because \(r\) and \(s\) are different variables, and no more operations apply. However, always check for any common factors or like terms that can be combined further. Simplification helps in solving equations and evaluating expressions more efficiently, thus making complex expressions easier to handle.