Problem 82 - Intermediate Algebra

Question

Simplify each numerical expression. $$ -2(-7+13)+6(-3-2) $$

Step-by-Step Solution

Answer

The simplified expression is -42.

Step 1: Simplify inside the parentheses

Let's start by simplifying the expressions inside each of the parentheses. First, simplify \(-7 + 13\):\(-7 + 13 = 6\).Now, simplify \(-3 - 2\):\(-3 - 2 = -5\).

Step 2: Substitute simplified expressions

Now substitute these simplified values back into the original expression:\(-2(6) + 6(-5)\).

Step 3: Perform multiplication

Carry out the multiplication step by step:First, multiply \(-2\) by \(6\):\(-2 \times 6 = -12\).Next, multiply \(6\) by \(-5\):\(6 \times -5 = -30\).

Step 4: Add the results of the multiplication

Now we add the results from the previous step together:\(-12 + (-30) = -12 - 30 = -42\).

Key Concepts

Order of Operations

When dealing with expressions like \(-2(-7+13)+6(-3-2)\), it's important to know the order in which you should perform mathematical operations. The order of operations helps you solve expressions in a consistent and standard manner. Here's the basic rule: remember the acronym PEMDAS.

• Parentheses: Do operations inside parentheses first.
• Exponents: Solve exponential expressions next.
• Multiplication and Division: Perform multiplication and division as they appear from left to right.
• Addition and Subtraction: Lastly, handle addition and subtraction from left to right.

In our original exercise, by following PEMDAS, we first simplify the expressions in the parentheses: \(-7 + 13 = 6\) and \(-3 - 2 = -5\). Then, we proceed with multiplication before finally adding the results of the multiplication. This ensures that we get the correct final result. Understanding and applying the order of operations is crucial when simplifying any type of expression.

Integer Arithmetic

Integer arithmetic involves operations such as addition, subtraction, multiplication, and division on whole numbers, which include positive numbers, negative numbers, and zero. Let's explore some important elements of integer arithmetic:

• Addition and Subtraction: When adding or subtracting integers, pay attention to signs. Adding two positive numbers or two negative numbers is straightforward: it’s like adding magnitudes and keeping the sign. For example, \(-3 - 2\) simply becomes more negative, resulting in \(-5\).
• Multiplication: When multiplying integers, remember the sign rule: a negative times a positive is a negative, and two negatives make a positive. So, in our exercise, \(-2 \times 6 = -12\) because we multiply a negative by a positive. Similarly, \(6 \times -5 = -30\).

Knowing these rules helps you solve expressions effectively and predictably, ensuring you don’t make errors with signs.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across terms inside parentheses. It states that: \(a(b+c) = ab + ac\). This property can be particularly useful when simplifying expressions. While the original exercise does not explicitly use this property, understanding it can enhance grasp of related expressions. Assume we had an expression like \(-2(x+y)\). Using the distributive property, it simplifies to \(-2x - 2y\). It's like distributing the \(-2\) across each term inside the parentheses. Remember, in multiplication, applying this property appropriately can simplify calculations. While our example operated straightforward multiplication on simplified terms (like \(-2(6)\) and \(6(-5)\)), thinking with the distributive property mindset can aid in understanding more complex expressions and algebraic manipulation.