Problem 120 - Prealgebra and Introductory Algebra

Question

Simplify. $$11-(-2)-6+10$$

Step-by-Step Solution

Answer

The simplified form of the expression \(11-(-2)-6+10\) is 17.

Step 1: Identify operations and numbers involved

Our expression is \(11-(-2)-6+10\). This expression includes four numbers: 11, -2, -6, and 10, and involves the operations of subtraction and addition.

Step 2: Apply rules for adding and subtracting integers

We need to remember that subtracting a number is the same as adding its opposite. So, we can read the expression in words as '11 plus 2 minus 6 plus 10'. This simplifies the operation and we can rewrite it as \(11+2-6+10\).

Step 3: Perform the operations

Perform the operations by respecting the order of operations. Since there are only additions and subtractions and they have the same precedence, do them from left to right: \(11+2-6+10 = 13-6+10 = 7+10 = 17\).

Key Concepts

Adding and Subtracting Integers

Understanding how to work with integers, which include both positive and negative numbers, is fundamental when simplifying expressions in algebra. Adding and subtracting integers isn't as straightforward as it is with only positive numbers, but the rules guiding these operations are simple and consistent.

When we add integers, if both numbers are positive or negative, we simply add their absolute values. For example, adding \(5 + 3\) gives us \(8\), as does adding \(-5 + (-3)\), which results in \(-8\). However, when adding a positive and a negative integer, the numbers effectively cancel each other out, up to the point of the larger absolute value. So, adding \(5 + (-3)\) would leave us with \(2\), as if we had subtracted \(3\) from \(5\).

Subtracting integers, on the other hand, can be thought of as adding the opposite. Specifically, when we subtract a negative number, like \(-2\), it's the same as adding its positive counterpart. This leads to less confusion when dealing with expressions that involve both operations. For instance, the subtraction of a negative number as in our problem, \(11 - (-2)\), simplifies to \(11 + 2\), making the process both easier to perform and to understand.

Order of Operations

The order of operations is crucial for solving any mathematical expression correctly. To avoid ambiguity and ensure consistency, mathematicians around the world adhere to a standard sequence when performing calculations, memorably expressed through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).

In the case of addition and subtraction, these operations are performed at the same precedence and are carried out from left to right, as they appear in the expression. This is why in our example expression \(11+2-6+10\), we added \(11\) and \(2\) first to get \(13\), then subtracted \(6\) to get \(7\), and finally, added \(10\) to arrive at the correct answer, \(17\).
Glancing at any expression, you might be tempted to jump right in and start calculating, but remember, always take a brief moment to recall PEMDAS, and it will guide your operations step by step to the right answer.

Basic Algebra

Basic algebra is the aspect of mathematics where we use letters and symbols to represent numbers and quantities in formulas and equations. It allows us to create general solutions that can work for a variety of cases, not just specific numbers. At its core, it's about finding unknown values or simplifying expressions.

In the problem \(11-(-2)-6+10\), we apply basic algebraic principles to simplify the expression. Just like we deal with numbers, algebra often requires us to simplify terms to make equations more manageable. Whether we're combining like terms, applying the distributive property, or balancing equations, the goal is always to break down the complexity into something we can easily understand and solve.
In many ways, simplifying this numerical expression is practicing algebra at its simplest form. As you advance in algebra, the operations will get more complex, but the foundational idea of making things simpler to solve remains the same. Always remember, behind every complex algebraic expression, there is a simpler form waiting to be revealed through the use of these fundamental operations and principles.