Problem 82 - Prealgebra

Question

In the following exercises, add. $$(a)7+5 \quad \text { ( b ) } 5+7$$

Step-by-Step Solution

Answer

Both 7 + 5 and 5 + 7 equal 12.

Step 1: Understand the Problem

The exercise requires adding two pairs of numbers. Arithmetic addition is to be used for both pairs.

Step 2: Add the Numbers for Part (a)

Take the first pair of numbers from part (a), which is 7 and 5. Add them together: \[7 + 5 = 12\]

Step 3: Add the Numbers for Part (b)

Take the second pair of numbers from part (b), which is 5 and 7. Add them together: \[ 5 + 7 = 12 \]

Step 4: Verify Commutative Property of Addition

Notice that in both parts, the sum is the same. This demonstrates the commutative property of addition where \(a + b = b + a\). Here, both \(7 + 5\) and \(5 + 7\) result in 12.

Key Concepts

commutative property

The commutative property of addition means that you can swap the numbers you're adding, and the result will be the same. This concept can be expressed as \( a + b = b + a \). In our exercise, we have 7 + 5 and 5 + 7.
When we add 7 + 5, we get 12. When we add 5 + 7, we also get 12. No matter the order of the numbers, the sum stays the same.

Understanding this property is important because it simplifies calculations and helps verify your answers. If you can swap the numbers and still get the right result, it's easier to check your work.
The commutative property isn't just for addition; it applies to multiplication too. However, it doesn't work for subtraction or division. Remembering that can help you avoid mistakes in more complex math problems.

addition operation

Addition is one of the basic operations in arithmetic. It involves combining two or more numbers to get a total. In our exercise, the numbers being added are 7 and 5.
The symbol for addition is \(+\), and it looks like this: \( a + b \). In our case, the equation is \( 7 + 5 \). When we add these numbers, we get 12.

Addition is cumulative, meaning the order in which you add numbers doesn't affect the sum. This brings us to the commutative property we saw earlier.
Practicing addition helps with more advanced math topics like algebra and calculus. It's also a handy skill in everyday life, like when you're adding up prices while shopping or keeping track of scores in a game.

problem-solving steps

Solving problems methodically helps ensure you understand each part of the process. Let's break down the steps we followed:

• Step 1: Understand the Problem
  First, identify what you need to do. In this case, it was adding two pairs of numbers.
• Step 2: Add the Numbers for Part (a)
  Take the first pair (7 and 5), and add them. Here, \( 7 + 5 = 12 \).
• Step 3: Add the Numbers for Part (b)
  Add the second pair (5 and 7), to get \( 5 + 7 = 12 \).
• Step 4: Verify Commutative Property of Addition
  Check if \( a + b = b + a \). Here, 7 + 5 and 5 + 7 both equal 12, confirming the property.

Following these steps ensures you understand the concepts and can solve similar problems confidently. Practice applying these steps to different sets of numbers to build your skills.