Problem 22 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 14+1 x 1 x 2-116 $$

Step-by-Step Solution

Answer

The simplified result is \(-100\).

Step 1: Identify the Operations

First, look at the mathematical expression given: \(14 + 1 \times 1 \times 2 - 116\). This expression involves addition, multiplication, and subtraction.

Step 2: Apply Order of Operations (Multiplication First)

According to the order of operations (PEMDAS/BODMAS), operations should be performed in this order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Hence, we first perform the multiplication: \(1 \times 1 \times 2 = 2\).

Step 3: Substitute the Result

Substitute the result from Step 2 back into the expression: \(14 + 2 - 116\).

Step 4: Perform Addition

Now, perform the addition operation: \(14 + 2 = 16\).

Step 5: Perform Subtraction

Finally, perform the subtraction operation: \(16 - 116 = -100\).

Step 6: Final Result

The simplified result of the expression \(14 + 1 \times 1 \times 2 - 116\) is \(-100\).

Key Concepts

PEMDAS

PEMDAS is a useful acronym that helps us remember the correct order of operations when solving mathematical expressions. PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. - **Parentheses**: Always solve expressions within parentheses first. They are like a "do this first" signal. However, in our exercise, there are no parentheses, so we move on to the next step.- **Exponents**: These are handled next, but aren't present in our exercise either.- **Multiplication and Division**: After exponents, tackle multiplication or division from left to right.- **Addition and Subtraction**: Similarly, these should be solved from left to right following multiplication and division.In the provided problem, we see the expression: \(14 + 1 \times 1 \times 2 - 116\). We begin by focusing on the multiplication \(1 \times 1 \times 2\) since it's the first operation needing attention. Understanding PEMDAS helps us tackle such expressions methodically and accurately.

simplifying expressions

Simplifying expressions involves performing mathematical operations in a systematic way to reduce an expression to its simplest form. The process makes complex problems easier to understand and solve by gradually breaking them down. In the exercise, the expression given is \(14 + 1 \times 1 \times 2 - 116\). Here's how simplification works step-by-step:- **Identify Operations**: First, pinpoint which operations are present—in this case, addition, multiplication, and subtraction.- **Apply PEMDAS**: Perform calculations based on the order dictated by PEMDAS. Here multiplication is the highest priority present, so calculate \(1 \times 1 \times 2 = 2\).- **Substitution**: With the result from multiplication, replace that part of the expression. This yields \(14 + 2 - 116\).Finally, resolve the new expression by performing the addition and subtraction in that sequence: \(14 + 2 = 16\) and then \(16 - 116 = -100\). The objective is to simplify the expression step-by-step until it cannot be simplified further.

elementary algebra

Elementary algebra is the foundation of algebra and includes the basic rules and simpler operations, such as addition, subtraction, multiplication, and division. It is essential in developing the ability to solve equations and understand more complex concepts later on. This exercise employs elementary algebra to simplify the expression \(14 + 1 \times 1 \times 2 - 116\). Here’s how elementary algebraic principles apply:- **Basic Operations**: The problem requires knowledge of fundamental arithmetic operations, integrated with algebraic reasoning.- **Variable Elimination**: While the problem doesn't explicitly include variables, part of algebra's essence is handling them. A simple number like the \(1 \times 1\) is akin to eliminating redundancy or a coefficient of an unseen variable.Elementary algebra simplifies expressions by using the rules we know, iterating through calculations step by step, checking work as you go, just as we solved for \(-100\) as the final result. This problem helps build the base required for tackling more advanced algebraic problems.