Problem 18 - College Algebra with Modeling and Visualization

Question

Evaluate by hand. $$ -2(16-3 \cdot 5) \div 2 $$

Step-by-Step Solution

Answer

-1

Step 1: Parentheses

First, handle the calculation inside the parentheses. Here, we have \(16 - 3 \cdot 5\). According to the order of operations, we perform the multiplication inside the brackets first: \(3 \cdot 5 = 15\). So, this becomes \(16 - 15\). Calculate \(16 - 15\) to get \(1\).

Step 2: Multiplication

Now substitute back into the expression: \(-2(1) \div 2\). Perform the multiplication \(-2 \cdot 1 = -2\).

Step 3: Division

After completing the multiplication, proceed with the division: \(-2 \div 2\). Perform the division to get \(-1\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operational symbols. These can be as simple as a single number or variable, or as complex as a combination involving multiple operations. In the given exercise, the algebraic expression is \(-2(16-3 \cdot 5) \div 2\) which involves a combination of numbers and operations.

• The numbers are \(-2, 16, 3, 5,\) and \(2\).
• The operations include multiplication \((\cdot)\), subtraction \((- )\), and division \((\div)\).

For solving such expressions, knowing the order of operations is crucial. This expression is particularly interesting because it includes parentheses, which indicates which operation to perform first.

Multiplication and Division

Multiplication and division are fundamental operations in algebra. They are performed after operations inside the parentheses and before addition and subtraction in an expression without parentheses.When dealing with expressions like the one in the exercise, it's essential to understand how multiplication and division interrelate. They are ranked equally in the hierarchy of operations, meaning you evaluate them from left to right as they appear.

• In the first step, within the parentheses, we encountered the multiplication: \(3 \cdot 5 = 15\).
• After solving the parentheses, the expression turned into \(-2 \cdot 1 \div 2\).
• Here, multiplication is done first, resulting in \(-2\).

Finally, division is performed, which will simplify our problem to completion. Thus, keen attention to detail is required to correctly execute these operations.

Evaluating Expressions

Evaluating expressions means finding the value of an expression when the values of the variables are known, or as is the case in this exercise, applying arithmetic operations systematically. Here's how to evaluate the given expression:Start with operations inside parentheses, simplify them, and then move outward. Follow the hierarchy of operations (often remembered by the acronym PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction), performing each operation step-by-step. In our exercise:

• First, the expression within the parentheses was simplified from \(16 - 3 \cdot 5\) to simply \(1\).
• Next, the expression was updated to \(-2(1) \div 2\).
• Following PEMDAS, multiplication was done first yielding \(-2\).
• Finally, division simplified the expression to \(-1\).

Thus with careful step-by-step simplification and adhering to operations' order, evaluating expressions becomes straightforward.