Problem 16 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. $$5(7+3)$$

Step-by-Step Solution

Answer

The equivalent expression using the Distributive Property is \(5 \times 7 + 5 \times 3\), and it evaluates to 50.

Step 1: Apply the Distributive Property

The Distributive Property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). Here, \(a = 5\), \(b = 7\), and \(c = 3\). Apply the property to obtain the expression: \(5 \times 7 + 5 \times 3\).

Step 2: Evaluate Each Term Separately

Calculate the value of each term from the expression obtained in Step 1: \(5 \times 7 = 35\) and \(5 \times 3 = 15\).

Step 3: Add the Results

Now add the results of the two products: \(35 + 15\).

Step 4: Calculate the Final Result

Add the numbers from Step 3 to get the final result: \(35 + 15 = 50\).

Key Concepts

Equivalent Expression

An equivalent expression is a different way of writing the same mathematical expression that produces the same result when evaluated. By using the Distributive Property in this task, we are creating an equivalent expression for the original one,
which is given as \(5(7+3)\). Instead of adding first and then multiplying, you can distribute the multiplication over addition. This means you multiply each term inside the parenthesis by the number outside.

• Start by identifying the parts: \(a = 5\), \(b = 7\), and \(c = 3\).
• The equivalent expression after using the Distributive Property becomes \(5 \times 7 + 5 \times 3\).

This not only shows a new perspective on solving problems but also adheres to mathematical properties that ensure the correctness and equality of expressions.

Evaluate Expression

Evaluating an expression involves calculating its value. It's similar to solving a problem where the numbers and operations determine an answer. For the expression derived from the Distributive Property
\(5 \times 7 + 5 \times 3\), we will find its value by doing the operations step by step.

• First, multiply: \(5 \times 7 = 35\)
• Second, multiply: \(5 \times 3 = 15\)
• Then, add the results: \(35 + 15\)

After these steps, the expression simplifies to 50. This shows the equivalency between the original and distributed expressions. Properly evaluating each term leads to an accurate and final solution.

Step-by-Step Solution

Understanding how to reach the final result is crucial, especially when breaking down expressions through the Distributive Property. Let's walk through the solution process step by step.First, dissect the expression using the Distributive Property. Here, you depict the operation of multiplying each internal term separately by the external multiplier, which was solved as \(5(7+3)\) becoming \(5 \times 7 + 5 \times 3\).Then, execute each multiplication:

• Calculate \(5 \times 7\), resulting in 35.
• Calculate \(5 \times 3\), resulting in 15.

Now, with both products separately found, you add them together. Adding the smaller results, \(35 + 15\), gives you the final result of 50. This step-by-step solution approach enables students to systematically solve similar problems by applying logical steps.