Problem 99 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. Simplify the expression in each exercise. $$ 2(-30-(-3))-3(6-9)+(-1)(1-15) $$

Step-by-Step Solution

Answer

-31

Step 1: Simplify Inside Parentheses/Brackets

Calculate the values inside each pair of parentheses first.\[2(-30-(-3))-3(6-9)+(-1)(1-15) \Rightarrow 2(-30+3)-3(-3)+(-1)(-14) \Rightarrow 2(-27)-3(-3)+(-1)(-14) \]

Step 2: Multiplication/Division

Perform the multiplication operations next.\[2(-27)-3(-3)+(-1)(-14) \Rightarrow -54+9+14\]

Step 3: Addition/Subtraction

Perform the final addition/subtraction operations from left to right.\[-54+9+14 \Rightarrow -45+14 \Rightarrow -31\]

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying algebraic expressions. This set of rules guides you on which procedures to perform first to accurately solve an expression. The acronym PEMDAS is often used to remember the order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the given exercise, the order of operations is applied: first the expressions within parentheses are resolved, then multiplication and division, and finally addition and subtraction. Following this rule ensures the correct simplification of the algebraic expression to reach the final solution of \( -31 \).

Parentheses in Algebra

Parentheses play a significant role in algebra as they dictate the priority of operations in an expression. When you see a problem with parentheses, it's like a signal that says 'solve me first!' This principle is demonstrated in our exercise, where we begin by simplifying the expressions within the parentheses.

As shown in the solution, \( -30 - (-3) \) simplifies to \( -27 \) by adding \( 3 \) (the double negative turns into a positive), and similarly for the other terms in parentheses. Misinterpreting these signs or neglecting the parentheses could lead to incorrect results, emphasizing their essential role in algebraic operations.

Basic Algebraic Operations

The foundation of algebra lies in basic operations such as addition, subtraction, multiplication, and division. In the context of algebra, these operations can involve whole numbers, variables, and parenthetical groups. After simplifying the expressions within the parentheses, we need to perform the multiplication or division operations.

In our exercise, proceeding after the parenthesis step, the multiplication operations are carried out, leading to \( -54 + 9 + 14 \). Finally, addition and subtraction are performed in sequence from left to right. These steps culminate in the simplification of the initial complex algebraic expression down to a single numerical value of \( -31 \), showing the importance of understanding and correctly applying basic algebraic operations.