Problem 41 - Algebra 1: Concepts and Skills

Question

Find the sum. $$6+10+(-6)$$

Step-by-Step Solution

Answer

The sum of \(6 + 10 + (-6)\) is \(10\).

Step 1: Addition of the first two numbers

Start by adding the first two numbers together, 6 and 10: \(6+10 = 16\)

Step 2: Addition of a Negative Number

Then, add the negative number. Remember, adding a negative is the same as subtracting. So, \(16 + (-6) = 16 - 6\)

Step 3: Perform the Subtraction

Subtract 6 from 16: \(16 - 6 = 10\)

Key Concepts

Addition of Integers

Integer addition is one of the basic building blocks of algebra. When we say 'integers', we're referring to whole numbers which can be either positive, negative, or zero. The addition of integers follows a straightforward rule: if the numbers have the same sign, you simply add their absolute values (the number without its sign) and keep the sign. If they have different signs, you subtract the smaller absolute value from the larger one, and the result takes the sign of the larger absolute value.

Let's look at a simple example: adding 6 and 10. Both integers are positive, so you combine their absolute values to get the sum of 16.

Adding Two Positive Integers

For two positive integers: \(6 + 10 = 16\).This result is also positive because both original numbers were positive.

Adding Negative Numbers

Adding negative numbers often confuses students, but it can be simplified with the right approach. When you're adding a negative number to a positive one, think of it as removing something from the positive amount. For example, if you have 16 apples, and you owe 6 apples to a friend, you would subtract 6 from your 16.

Understanding Negative Addition

When encountering a problem like \(16 + (-6)\), remember that the negative sign in front of the 6 indicates that you're taking away 6 from 16. Mathematically, it is equivalent to subtraction: \(16 + (-6) = 16 - 6\).By using this approach, you'll avoid common mistakes and make the concept of adding negatives more intuitive.

Subtraction

Subtraction is another fundamental operation in algebra, closely related to addition. Essentially, subtraction is the process of taking a number away from another number. It's easy to understand when dealing with positive numbers, but it gets a bit more intricate with negatives involved.

Performing Basic Subtraction

Once you have simplified the initial step of adding integers, any resulting subtraction should follow naturally. If we continue with the example we have, we subtract 6 from 16, simply because we're taking away part of a quantity. The calculation \(16 - 6 = 10\) yields the final result of our problem. The process shows that adding a negative number (which is the same as subtracting) doesn't have to be complicated—it's just about taking something away from a larger amount.