Problem 53 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each series of additions and subtractions. $$9-8+3-7$$

Step-by-Step Solution

Answer

-3

Step 1: Begin with the first operation

We start from the left and perform the first operation, which is subtraction: \(9 - 8 = 1\). So the expression now becomes \(1+3-7\).

Step 2: Execute the second operation

Next perform the addition operation: \(1 + 3 = 4\). So the expression now simplifies to \(4-7\).

Step 3: Carry out the last operation

Lastly, perform the subtraction operation: \(4 - 7 = -3\)

Key Concepts

Addition and Subtraction

Addition and subtraction are fundamental arithmetic operations that help us calculate and simplify expressions. In algebra, these operations are often used to combine or simplify numbers or terms. When simplifying a series, such as the one in this exercise — involving both addition and subtraction — the order in which you perform these operations greatly matters.
We work from left to right because this respects the order of operations, sometimes referred to as PEMDAS/BODMAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In the given problem, you have the expression: 9 - 8 + 3 - 7. Using the left-to-right rule:

• First, subtract 8 from 9, which gives us 1.
• Then, add 3 to the result, getting 4.
• Finally, subtract 7 from this result to arrive at the final answer, -3.

This careful sequence helps us ensure accuracy.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. They form the basis for nearly all of arithmetic and algebra.
Each operation has its rules and methods for application. For this exercise, we focus on addition and subtraction.
Addition is the operation of combining numbers to determine their total or sum. When you see a plus sign, you know the numbers on either side should be added.
Subtraction, on the other hand, determines the difference between numbers by taking away the second number from the first. The minus sign indicates this operation.
In our example, these operations work together to refine the expression step-by-step:

• Subtracting 8 from 9 simplifies the expression initially.
• Adding 3 helps to further define the result.
• Lastly, subtracting 7 gives us the final outcome.

Understanding how these operations interact and their specific roles helps simplify complex expressions.

Simplification Process

The simplification process in algebra involves reducing expressions into simpler or more manageable forms without changing their value. It often includes combining like terms and performing arithmetic operations in the correct order.
The goal is to make expressions easier to understand and solve. When we simplify, we often eliminate unnecessary numbers or terms, streamline calculations, and minimize errors.
In our exercise, the simplification takes place as follows:

• We start with the expression 9 - 8 + 3 - 7.
• The first step reduces 9 - 8 to 1.
• Next, by adding 3, we further simplify this to a cleaner 4.
• Finally, subtracting 7 from 4 yields -3, our simplest form for this expression.

This process is essential in algebra as it efficiently condenses information, making equations and expressions more understandable.