Problem 36 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-10-1+16$$

Step-by-Step Solution

Answer

The simplified form of the expression is 5.

Step 1: Rewrite Subtractions as Additions

First, let's change each subtraction in the expression to the addition of the opposite. The expression \(-10 - 1 + 16\) has a subtraction of \(-1\). We can write this as \(-10 + (-1) + 16\). This makes adding easier.

Step 2: Add from Left to Right

Now that we have an expression of only additions, we can add the numbers from left to right. Start with \(-10 + (-1)\). This equals \(-11\), as adding a negative is like subtracting the absolute value.

Step 3: Add the Final Number

Finally, add \(-11\) to the remaining number in the expression, which is \(16\). Thus, \(-11 + 16\) equals \(5\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental aspect of algebra that involves reducing them to their simplest form. This process generally requires following a series of steps to either combine like terms or eliminate unnecessary complexity.
Before simplifying a mathematical expression, it's important to identify the operations involved, like addition or subtraction.
Here are some tips to help you simplify expressions more effectively:

• Rewrite subtractions as additions, which can make the expression easier to handle.
• Combine like terms by performing operations in a left-to-right sequence.

In our specific exercise, simplifying involves converting subtractions to additions and then performing the operations from left to right to reach the simplest form. After these steps, achieving the result becomes straightforward and logical.

Addition of Integers

Adding integers is a fundamental mathematical skill that involves finding the total or sum of values. Integers include whole numbers and their opposites, such as negative numbers.
When performing addition with integers, remember:

• If both integers are positive, their sum is simply the total of their magnitudes.


• If both are negative, the sum is also negative, but you add their absolute values and then apply the negative sign back.


• For one positive and one negative integer, the sum will have the sign of the integer with the largest absolute value, using subtraction to find the result.

In the given exercise, we see these rules applied as \(-10 + (-1)\) involves both negative integers, leading to \(-11\). This understanding ensures clarity and accuracy when working with integer sums.

Subtraction as Addition of Opposites

Subtraction in mathematics can sometimes be confusing, but viewing it as addition of opposites makes it easier to understand. This concept relies on transforming a subtraction operation into an addition operation by adding the negative of a number.
For example, instead of subtracting 1, you add -1. This effectively changes the operation but not the outcome, making calculations simpler.
To apply this technique:

• Change the subtraction sign to an addition sign.


• Switch the number that follows to its opposite. For instance, instead of \(5 - 3\), consider \(5 + (-3)\).

In our exercise, the expression \(-10 - 1 + 16\) transforms to \(-10 + (-1) + 16\). This makes the sequence of operations more consistent, especially when reading from left to right. By conceptualizing subtraction as addition of opposites, it becomes much easier to manage and solve mathematical expressions.