Problem 77 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 19+2(-3+8) $$

Step-by-Step Solution

Answer

29

Step 1: Identify the Expression

The given expression to simplify is 19 + 2(-3 + 8)

Step 2: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses: -3 + 8 = 5 So the expression now is 19 + 2(5).

Step 3: Multiply

Next, perform the multiplication: 2(5) = 10. Now the expression is 19 + 10.

Step 4: Add

Finally, add the remaining numbers: 19 + 10 = 29.

Key Concepts

Order of Operations

The order of operations is a key mathematical principle that dictates the correct sequence in which to solve different parts of an expression. This sequence is often remembered through the acronym PEMDAS, which stands for:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

When dealing with any mathematical expression, always start by simplifying the parts inside the parentheses. Then, move on to any exponents. Next, perform any multiplication or division in the order they appear from left to right. Finally, carry out any addition or subtraction in the order they appear from left to right.

In the given exercise, following the order of operations ensures we correctly simplify the expression. We first handle the parentheses, followed by the multiplication, and finally, the addition.

Parentheses in Algebra

Parentheses are crucial in algebra as they indicate which operations should be performed first. When you see an expression with parentheses, always focus on simplifying what’s inside them before addressing any other part of the expression.

For instance, let's look at the given expression: 19 + 2(-3 + 8). Here, the part inside the parentheses is -3 + 8.

According to the order of operations, we first simplify -3 + 8 to get 5. Now the expression looks like this: 19 + 2(5). By dealing with the parentheses first, we simplify our task and avoid mistakes in further calculations. Without simplifying inside the parentheses first, we could miscalculate the entire expression.

Basic Arithmetic Operations

Understanding basic arithmetic operations such as addition, subtraction, multiplication, and division is fundamental to simplifying algebraic expressions. Let's break down each step in the given problem:

• Addition: Combining two numbers into a single sum, such as 19 + 10.
• Subtraction: Taking one number away from another, which was used when we simplified inside the parentheses -3 + 8.
• Multiplication: Scaling a number by another number. In our exercise, we multiplied 2 by the simplified result of the parentheses to get 2(5) = 10.

After performing each of the operations step-by-step, you correctly arrive at the final answer: 29. Knowing how and when to apply these operations is key to mastering algebra and other areas of math.