Problem 36 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(5(7+8 y)\)

Step-by-Step Solution

Answer

The simplified expression is \(35 + 40y\).

Step 1: Applying the Distributive Property

The distributive property tells us that multiplying a number by a sum is the same as multiplying each addend individually and then adding the products. For the expression \(5(7 + 8y)\), distribute 5 across the terms inside the parentheses: \[5 \cdot 7 + 5 \cdot 8y\].

Step 2: Calculate the Products

Now, calculate each product from Step 1. First, multiply 5 by 7: \(5 \times 7 = 35\). Then, multiply 5 by \(8y\): \(5 \times 8y = 40y\).

Step 3: Write the Simplified Expression

Combine the results of the products from Step 2 into a single expression. The expression is \(35 + 40y\). This is the simplified form of the original expression using the distributive property.

Key Concepts

Simplifying Algebraic Expressions

Understanding how to simplify algebraic expressions is crucial in making math problems more manageable. It involves breaking down complex expressions into simpler, more digestible forms. The idea is to transform expressions into their most compact and efficient version. This often includes removing parentheses by distributing coefficients, combining like terms, and reducing expressions to the fewest terms possible.

In the given problem, simplification starts with using the distributive property: taking a number outside of parentheses and applying it to each term inside the parentheses. Once that's done, you calculate the individual products (as seen in steps). Combining these results, like we combined the two products to form the expression \(35 + 40y\), is another key step in simplification. Look out for opportunities to merge terms when they share a common factor or variable.

Multiplication in Algebra

Multiplication in algebra isn't just about numbers; it can also involve variables. When you have an expression like \(5(7 + 8y)\), you're multiplying a number (5) with both a constant (7) and a variable term (8y). This brings us back to an important algebraic principle: you can distribute multiplication across terms inside parentheses.

• Multiply the constant, like multiplying 5 by 7 to get 35.
• Next, multiply the coefficient with the variable term, such as multiplying 5 by 8y, resulting in 40y.

This step is crucial because it allows you to simplify expressions whilst maintaining equality. Moreover, remember the importance of maintaining variable integrity—only terms with the same variable can be directly combined. This ensures your final simplified expression remains accurate and true to the original form.

Algebraic Properties

Algebraic properties are foundational rules that make dealing with algebraic expressions easier. One of these fundamental properties is the distributive property which states that \(a(b + c) = ab + ac\). This property allows you to remove parentheses by distributing a multiplier across each term within the parentheses.

Another important aspect is understanding that operations follow specific laws, like associativity and commutativity. However, in the context of this exercise, the main focus is using the distributive property effectively. It's the principle that helps us break down expressions like \(5(7 + 8y)\) into \(5 \cdot 7 + 5 \cdot 8y\).

Consistency in applying these algebraic rules ensures that your simplified expressions are both correct and as reduced as possible, paving the way for more complex problem-solving without losing accuracy. Understanding these properties helps you recognize patterns and simplify calculations in your mathematical journey.