Problem 116 - Elementary Algebra

Question

Simplify. $$ (2 \cdot 8-62) 2-10273-(2(-5) 3-7) $$

Step-by-Step Solution

Answer

The simplified expression is -10328.

Step 1: Initial Expression

Write the initial expression to be simplified: \[(2 \cdot 8 - 62) \cdot 2 - 10273 - (2(-5) \cdot 3 - 7)\]

Step 1: Simplify Inside the Parentheses

First, simplify inside each set of parentheses.1. Calculate \(2 \cdot 8 = 16\).2. Replace and simplify \((16 - 62) = -46\).3. For the second set of parentheses, calculate \(2(-5) = -10\).4. Then, \(-10 \cdot 3 = -30\).5. So, replace and simplify \(-30 - 7 = -37\).Now the expression is:\[(-46) \cdot 2 - 10273 - (-37)\]

Step 2: Multiply

Multiply the result from the first parentheses by 2: 1. Calculate \(-46 \cdot 2 = -92\).The expression becomes:\[-92 - 10273 - (-37)\]

Step 3: Eliminate Double Negatives

Eliminate the double negative by converting \(-(-37)\) to \(+37\):\[-92 - 10273 + 37\]

Step 4: Simplify the Expression

Perform the arithmetic step by step:1. \(-92 - 10273 = -10365\).2. Then add 37: \(-10365 + 37 = -10328\).

Step 6: Final Answer

The simplified expression is:\[-10328\]

Key Concepts

Order of Operations

Whenever you simplify expressions in mathematics, it's very important to follow the order of operations. This is essential to ensure that everyone solves the expression in the same way and arrives at the correct answer. The order of operations, which is often remembered using the acronym PEMDAS, helps us remember which calculations to perform first.

• **Parentheses** (P): Do any operations inside parentheses or brackets first.
• **Exponents** (E): Next, calculate all exponential terms.
• **Multiplication and Division** (MD): These come next and should be performed from left to right as they appear.
• **Addition and Subtraction** (AS): Finally, perform addition and subtraction, again from left to right.

In our example problem, the order of operations is crucial. You start by simplifying expressions inside the parentheses. Then move on to multiplication, and finally tackle addition and subtraction. Skipping any of these steps or doing them out of order could lead to an incorrect result.

Parentheses in Algebra

Parentheses play a pivotal role in algebraic expressions. They indicate which operations should be performed first, regardless of where they are in the expression. Understanding and using parentheses correctly can greatly affect the outcome of your simplification. Here are some key points:

• Operations inside parentheses should be done before anything outside of them.
• Parentheses can also help in clarifying expressions and making them easier to read and solve.
• If there are nested parentheses, work from the innermost to the outermost layer.

In the given exercise, simplifying expressions in the parentheses was the first step. This involved operations like multiplication and subtraction within the parentheses, helping to simplify the expression for further operations. Without handling these first, the rest of the simplification process would lead to errors.

Negative Numbers Operations

Dealing with negative numbers can often be tricky, yet it's an essential skill in algebra and arithmetic. Here are some tips on how to manage negative numbers effectively:

• **Multiplying Negative Numbers:** When you multiply two negative numbers, the result is positive. Conversely, multiplying a positive number by a negative number gives a negative result.
• **Subtracting Negative Numbers:** This can be seen as adding the opposite. For example, subtracting \(-7\) is the same as adding 7. This rule was followed in our problem when we eliminated double negatives by converting \(-(-37)\) to \(+37\).
• **Adding Negative Numbers:** This is akin to regular addition, but you move in the negative direction on the number line.

In our example, understanding how to handle negative numbers was critical. Multiplying \(-5\) by \(3\) within the parentheses initially produced \(-30\), and then careful management of negatives ensured the correct simplified result.