Problem 66 - Prealgebra and Introductory Algebra

Question

Use the given property of addition to complete the statement. The Associative Property of Addition $$-11+(6+9)=(?+6)+9$$

Step-by-Step Solution

Answer

There seems to be an error in the application of the Associative Property of Addition. The correct calculation should be \(-11+15 = 4\), not \(-11+15 = -26\). Hence, the correct equation should be \(-11+(6+9)=4=(4+6)+9\).

Step 1: Use the Associative Property of Addition

With the Associative Property of addition, does not matter how the numbers are grouped. Hence, the equation could be written as, \(-11+(6+9)=(-11+6)+9\)

Step 2: Solve for the equation in both sides

On the left-hand side, calculate the sum of 6 and 9 first. So we have \(-11+(6+9)=-11+15=-26\). On the right-hand side, calculate the sum of -11 and 6 first which equals -5; then, add 9 to -5. Hence, \((-11+6)+9=-5+9=4\).

Step 3: Compare and Identify the Error

Both side of the equation do not equal to each other (-26 ≠ 4), indicating that there is an error made in the process. The exercise seems to misapply the Associative Property of Addition. In context, the incorrect application is \(-11+15 = -26\). The correct calculation is \(-11+15 = 4\).

Key Concepts

Understanding Addition Properties

Addition properties are foundational concepts in mathematics that make calculations more manageable and flexible.
One of these essential properties is the **Associative Property of Addition**. This property states that the way you group numbers when adding them does not alter the sum.

For example:

• The expression \(a + (b + c)\) is equivalent to \((a + b) + c\).
• This means you can move parentheses around, and the result will remain the same.

By properly using this property, you can simplify complex calculations and avoid unnecessary steps.

In our exercise, the associative property allows us to regroup and rearrange the numbers to make calculations simpler without changing the outcome. Remembering this property is crucial, especially in solving algebraic expressions and avoiding errors.

Analyzing Algebraic Expressions

Algebraic expressions involve numbers combined with variables and mathematical operations.
In this exercise, even though specific values were used, the principle of manipulating algebraic expressions remains the same. Understanding this concept enables you to solve various problems effectively.

When dealing with algebraic expressions, it is essential to keep in mind:

• Identify and separate the terms involved in the calculation.
• Apply relevant properties, like the Associative Property, to simplify the expression.
• Ensure consistency in operations to arrive at an accurate result.

These strategies will help you to manipulate and solve algebraic expressions correctly.
By practicing these steps with different problems, you'll develop a stronger grasp of how algebraic expressions work.

Correcting Mathematical Errors

Making mistakes is a natural part of learning mathematics. However, recognizing and correcting them is crucial for solid understanding.
In the exercise, an error was initially made by misapplying the Associative Property of Addition.

Here's how you can prevent and correct mathematical errors:

• Always double-check calculations, especially when dealing with properties.
• Verify both sides of an equation to ensure they match.
• Take a step back to understand the concept if results seem inconsistent.

In this instance, realizing that \( -11 + 15 \) did not correctly equal \( -26 \) led to the understanding that the actual sum should be \(4\).
We must diligently review our work to avoid repetitive mistakes.
By practicing these measures, you will develop a stronger mathematical intuition and enhance your problem-solving skills.