Problem 87 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ 8(3 r+4 s-5 y) $$

Step-by-Step Solution

Answer

24r + 32s - 40y

Step 1 - Understand the Distributive Property

The distributive property states that for any numbers a, b, and c: $$a(b + c) = ab + ac$$ This property allows us to distribute a multiplied value to each term inside the parentheses.

Step 2 - Identify the Terms

In the given expression, identify the number outside the parentheses and the terms inside the parentheses. In this case, 8 is outside the parentheses and $$3r + 4s - 5y$$ are the terms inside the parentheses.

Step 3 - Apply the Distributive Property

Multiply each term inside the parentheses by the number outside. Calculate the following: 8 * 3r = 24r 8 * 4s = 32s 8 * (-5y) = -40y

Step 4 - Combine the Results

Combine the results from the previous step: 24r + 32s - 40y. This is the expanded form of the original expression.

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations where you combine equal groups. In algebra, multiplication often involves combining numbers and variables. When you see an expression like \(a(b + c)\), you need to multiply (or distribute) \(a\) to both \(b\) and \(c\). This is what the distributive property is all about. For example, if you have the expression \(8(3r + 4s - 5y)\), you multiply 8 by each term inside the parentheses separately. This results in \(8 * 3r + 8 * 4s + 8 * (-5y)\).

Algebraic Expressions

An algebraic expression combines numbers, variables, and operators (like addition and multiplication). In the expression \(8(3r + 4s - 5y)\), we have numbers (8, 3, 4, -5) and variables (r, s, y). The parentheses indicate that we should apply the distributive property. Each operation inside the parentheses involves variables, making it an algebraic expression. Remember that the goal is often to simplify these expressions to make them easier to work with in other math problems.

Combining Like Terms

Once you've distributed a number across an algebraic expression, it's often helpful to combine like terms. Like terms are terms that contain the same variables raised to the same power. For example, in \(24r + 32s - 40y\), each term is different (one with r, one with s, one with y), so you can't combine them further. However, if you had something like \(24r + 10r\), you could combine these to get \(34r\). This simplification process is crucial in making expressions easier to understand and solve.

Step-by-Step Solution

Using a step-by-step method is crucial in understanding and solving math problems. Let's go through each step for the expression \(8(3r + 4s - 5y)\):

- **Step 1:** Understand the distributive property. This property shows how to multiply a single term by each term inside parentheses.
- **Step 2:** Identify the terms outside and inside the parentheses. Here, 8 is outside, and \(3r + 4s - 5y\) are inside.
- **Step 3:** Apply the distributive property. Multiply 8 by each term inside: \(8 * 3r = 24r, 8 * 4s = 32s, 8 * (-5y) = -40y\).
- **Step 4:** Combine the results into the final expression: \(24r + 32s - 40y\).

By breaking down the problem into these smaller, manageable steps, you can solve even complex expressions more easily.