Problem 64 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -17-32-(-85)-51 $$

Step-by-Step Solution

Answer

The result of the operation is -15.

Step 1: Identify Like Terms

Inspect the given expression: \(-17 - 32 - (-85) - 51\)First, identify and separate each term in the expression.

Step 2: Simplify Negative Signs

Notice the expression has a double negative: \(-(-85)\).Simplifying this gives \(+85\). So, the expression becomes: \(-17 - 32 + 85 - 51\).

Step 3: Combine Negative Terms

Add together all the negative numbers: \(-17 - 32 - 51 = -100\).

Step 4: Add Result with Positive Term

Combine the sum of the negative terms \(-100\) with the positive term \(+85\): \(-100 + 85 = -15\).

Key Concepts

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression easier to understand and work with. This involves removing unnecessary components and performing operations like combining terms or using properties of arithmetic to create a simpler expression.
To simplify expressions effectively, follow these steps:

• Identify the terms involved in the expression.
• Look for ways to make the expression easier, like changing double negatives to positives.
• Use basic arithmetic operations such as addition and subtraction.

In our example, the initial expression \(-17-32-(-85)-51\) became easier when we simplified the double negative from \(-(-85)\) to \(+85\). This change made the expression easier to handle and clearer to solve.

Combining Like Terms

Combining like terms is an essential skill in simplifying expressions. Like terms are those that share the same variable or are simply numerical values. In our exercise, all terms are integers, which makes them candidates for combining directly.
To combine like terms:

• Identify terms that can be combined. They must be either numbers or share the same variable.
• Use addition or subtraction to merge these terms into a single term.

In our given expression, we identified the integers: \(-17\), \(-32\), \(85\), and \(-51\). We first combined the negative numbers \(-17\), \(-32\), and \(-51\) to get \(-100\), before adding this to the positive number \(85\). This streamlined the expression into fewer, simpler terms.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus (-) sign. They are critical in many mathematical operations, and understanding how they behave during arithmetic processes is crucial.
Dealing with negative numbers involves several key aspects:

• Addition of a negative number is the same as subtraction: \(a + (-b) = a - b\).
• Subtracting a negative number is the same as addition due to double negation: \(a - (-b) = a + b\).
• Negative numbers combined with each other result in more negative values. For example, \(-3 + (-2) = -5\).

In our example, managing the negative terms was crucial. The expression contained \(-17 - 32 - 51\), all negative, which added together equaled \(-100\). It’s important to recognize how the expression \(-(-85)\) converts into a positive 85.

Positive Numbers

Positive numbers are greater than zero and do not carry any sign in basic mathematical notation. They are simple to work with as they follow standard rules of arithmetic without requiring special considerations such as those with negative numbers.
Here are a few points about positive numbers:

• Adding a positive number to another number increases the original value.
• Subtracting a positive number decreases the original number.
• In mixed expressions, prioritize combining positive numbers where possible for simpler calculations.

In our exercise, the positive number we worked with was \(+85\) which resulted from simplifying \(-(-85)\). Following the combination with the total negative sum \(-100\), we arrived at the final result of \(-15\). Recognizing and manipulating positive numbers within expressions is key to finding accurate solutions.