Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each series of additions and subtractions. $$-19-8-(-6)+(-21)$$

Step-by-Step Solution

Answer

-42

Step 1: Identify Positive and Negative Numbers

In the expression \(-19-8-(-6)+(-21)\), \(-19\), \(-8\), and \(-21\) are negative numbers whereas \(-6\) is subtracted which makes it a positive number (since subtracting a negative number is same as adding the positive number).

Step 2: Rearranging of the terms

Rearrange the terms to group positive /\(-6\)/ and negatives \(-19\), \(-8\), and \(-21\): \[ -19 - 8 - 21 + 6 \]

Step 3: Summation of Positive Numbers

Add together positive numbers. In this case we only have one positive number, \(6\)

Step 4: Summation of the Negative Numbers

Add together negative numbers(\(-19\), \(-8\), \(-21\)). The sum is \(-48\)

Step 5: Final Summation

Now add the sum of positive numbers and negative numbers together. The sum should be \( -48+6= -42\) .

Key Concepts

Positive and Negative Numbers in Algebra

Understanding the distinction between positive and negative numbers is foundational in algebra. When we see a number without a sign, it is assumed to be positive. Negative numbers, on the other hand, are indicated by a preceding minus (-) sign.

For instance, in the expression \( -19-8-(-6)+(-21) \), \( -19 \), \( -8 \) and \( -21 \) are readily identified as negative. But \( -(-6) \) might be tricky for some. This is where the rules for arithmetic with negative numbers come in handy: subtracting a negative is the same as adding its positive counterpart. So \( -(-6) \) becomes \( +6 \), turning it into a positive number.

• Notice that adding positive and negative numbers reflects moving along a number line in opposite directions.
• When adding a negative number, you move left from zero on the number line, indicating a decrease.
• Adding a positive number means moving right, which shows an increase.

If we grasp this concept, it sets the stage for more complex arithmetic, such as dealing with algebraic expressions involving variables.

Rearranging Terms to Simplify Algebraic Expressions

Rearranging terms in an algebraic expression can be likened to organizing a cluttered room - grouping similar items makes it easier to see what you have. We do this by moving terms around, keeping in mind the law of commutativity, which allows us to add numbers in any order.

Commute with Caution

When we rearrange, it is essential to carry the sign in front of a number along with it. For example, in the given expression \( -19-8-(-6)+(-21) \), regrouping the negative numbers \( -19 \), \( -8 \) and \( -21 \) together highlights the total amount of 'debt', whereas \( +6 \) represents a 'credit'.

By rearranging, we simplify the task of addition and subtraction:

• We usually group like terms (all the negative or positive numbers).
• This consolidation allows us to deal with fewer terms, making our calculations simpler and less prone to error.

Remember, rearranging is merely a strategy to facilitate easier computation; it doesn't change the value of the expression.

Summation in Algebra

Summation in algebra refers to adding a sequence of numbers (terms). It's important to understand that while this concept might seem simple with small numbers, it lays the ground for handling complex equations and larger sums.

In our example, we need to perform two summations: one for positive and another for negative numbers. After rearranging, we sum the negative numbers \( -19 \), \( -8 \) and \( -21 \) to find the total negative sum, which is \( -48 \). The positive sum, in this case, is straightforward as we have a single positive term, \( 6 \).

• The final summation combines both results, giving us \( -48 + 6 = -42 \).
• It's essential to keep track of the signs when performing a summation to avoid confusion.
• Consistently applying the rules for arithmetic operations will yield correct results.

By simplifying algebraic expressions through summation, we break down complex problems into more manageable steps, allowing for clearer understanding and solving.