Problem 123 - Prealgebra and Introductory Algebra

Question

Simplify. $$-13+9-(-10)-4$$

Step-by-Step Solution

Answer

The simplified form of the mathematical expression \(-13+9-(-10)-4\) is \(2\).

Step 1: Understanding the Exercise

First, analyze the given expression \(-13+9-(-10)-4\). Always remember that subtracting a negative number is the same as adding a positive equivalent of that number.

Step 2: Remove Parentheses

The parenthesis expression -(-10) can be simplified as: \(-(-10) = 10\). Replacing this in the expression, we get: \(-13+9+10-4\).

Step 3: Carry out the Operation

Calculate the value step by step. First, add -13 and 9 to get -4. Then add 10 to -4 to get 6. Finally, subtract 4 out from 6 to get 2. So, \(-13+9+10-4 = 2\).

Key Concepts

Integer Operations

Integer operations are fundamental when simplifying mathematical expressions. Integers include positive numbers, negative numbers, and zero. Understanding how to add, subtract, multiply, and divide integers is crucial for solving problems accurately.

• Addition: When adding two integers with the same sign, add their absolute values and keep the sign. For example, \(-2 + (-3) = -5\).
• Subtraction: To subtract integers, add the opposite. Subtracting \(-5 - 2\) is the same as \(-5 + (-2)\).
• Multiplication and Division: The product or quotient of two integers with the same sign is positive, and with different signs is negative. For example, \(-3\times 4 = -12\) and \(-12 \div (-3) = 4\).

Using these operations correctly helps in breaking down and solving complex expressions efficiently.

Negative Numbers

Handling negative numbers can sometimes be tricky, but they follow simple rules. Negative numbers are those less than zero, typically represented with a minus sign (-). When dealing with the original expression \(-13+9-(-10)-4\), keeping the negative sign rules in mind helps.

• Adding a negative is like subtracting a positive: \(9 + (-3) = 9 - 3\).
• Subtracting a negative is like adding a positive: \(-3 - (-10) = -3 + 10\).
• Two negative signs in succession (as in \(-(-10)\)) cancel out to create a positive number.

Understanding these fundamentals makes transitioning through expressions with negative numbers more manageable.

Parentheses in Math

Parentheses are a powerful tool in math to indicate which operations should be carried out first within an expression. In the exercise \(-13+9-(-10)-4\), the parentheses around \(-10\) signal an operation that needs consideration. Here, they direct us to deal with the \(-(-10)\) first.

• The primary rule is to simplify inside parentheses first, following the rest of the order of operations.
• They can also indicate multiplication when placed adjacent to another term, but in our example, they clarify the double negative operation.
• Removing parentheses is often the first step in simplifying an expression, ultimately aiding clarity in further calculations.

With the right approach to parentheses, you can simplify what's inside before tackling other parts of an expression.

Order of Operations

The order of operations is a set of rules to ensure everyone calculates a mathematical expression the same way. The mnemonic PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) can help remember this order.

• Start with any calculations inside parentheses.
• Handle exponents next, although there are none in our example.
• Perform all multiplication and division from left to right.
• Finally, complete addition and subtraction from left to right.

In the case of \(-13+9-\left(-10\right)-4\), after handling the parentheses, you move left to right with additions and subtractions: First \(-13 + 9 = -4\), then \(-4 + 10 = 6\), and lastly \(6 - 4 = 2\). This order is essential for solving expressions correctly.