Problem 15 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 1+1 x 2-1 x $$

Step-by-Step Solution

Answer

The simplified expression is \(3 - x\).

Step 1: Recognize the Expression

The given mathematical expression is \(1 + 1 \times 2 - 1 \times x\). It includes addition, subtraction, and multiplication operations.

Step 2: Apply PEMDAS/BODMAS Rules

According to the order of operations (PEMDAS/BODMAS), multiplication should be performed before addition and subtraction. So, calculate \(1 \times 2\) and \(-1 \times x\) first.

Step 3: Perform Multiplication

Calculate \(1 \times 2\) resulting in 2. Now, calculate \(-1 \times x\) resulting in \(-x\). The expression now becomes \(1 + 2 - x\).

Step 4: Perform Addition and Subtraction

Next, perform the operations from left to right. First, do the addition: \(1 + 2 = 3\). Then, subtract \(x\) from \(3\) to get \(3 - x\).

Key Concepts

Order of Operations

In arithmetic and algebra, the order of operations is essential to performing calculations accurately. This concept dictates the sequence in which mathematical operations should be carried out to ensure that everyone reads and solves expressions in the same way.
For the expression given, knowing the order in which to perform operations is key. The order of operations can be remembered by the acronym PEMDAS/BODMAS, which guides us through this process.

Simplification

Simplification in algebra refers to the process of making expressions easier to understand or solve. By combining like terms and eliminating unnecessary components, we make algebraic expressions more manageable.
In the original exercise, the goal is to simplify the expression by systematically applying the operations according to the rules. Simplifying expressions not only helps in solving them quickly but also assists in identifying equivalent expressions during comparisons.

PEMDAS/BODMAS

PEMDAS/BODMAS is a helpful acronym to remember the order of operations:

• P/B: Parentheses/Brackets
• E/O: Exponents/Orders (such as squares and square roots)
• MD/DM: Multiplication and Division (from left to right)
• AS: Addition and Subtraction (from left to right)

In the step-by-step solution of the problem, these rules guide us to process multiplication before moving on to addition and subtraction. This ensures the expression's value remains consistent and correct as it is simplified.

Mathematical Operations

Mathematical operations - such as addition, subtraction, multiplication, and division - form the foundation of algebra. Each operation performs a specific calculation and impacts the others based on the order they are performed.
In the given problem, learning to correctly apply multiplication before dealing with addition and subtraction is crucial. By handling each of these operations carefully and in the right order, you can find the correct simplified expression, in this case, leading to the final solution of \(3 - x\).