Problem 85 - Algebra for College Students

Question

Simplify each numerical expression. $$7(6.2-7.1)-6(-1.4-2.9)$$

Step-by-Step Solution

Answer

The simplified expression is 19.5.

Step 1: Simplify Inside the Parentheses

First, simplify the expressions inside each set of parentheses separately. For the first set, calculate \(6.2 - 7.1\), which equals \(-0.9\). For the second set, calculate \(-1.4 - 2.9\), which equals \(-4.3\).

Step 2: Apply the Multiplication

Next, multiply the numbers outside the parentheses by the simplified results inside the parentheses. Calculate \(7 \times -0.9\), which equals \(-6.3\). Then calculate \(-6 \times -4.3\), which equals \(25.8\).

Step 3: Perform the Subtraction

Finally, subtract the results from the previous step: \(-6.3 - 25.8\). This simplifies to \(25.8 - 6.3 = 19.5\). Remember that subtracting a negative is the same as adding the positive.

Key Concepts

Simplifying Expressions

When dealing with mathematical expressions, simplifying them is an essential skill. The goal is to reduce the expression to its simplest form. This often involves performing operations within parentheses first, in accordance with the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Simplifying involves:

• Breaking down complex expressions
• Reducing unnecessary complexity
• Ensuring accuracy of results

In our original exercise, simplifying was performed by calculating within the parentheses first. This simplifies the overall expression and prepares it for further operations. By handling parentheses first, we maintain clarity and precision. Always remember to work from the inside out when simplifying.

Multiplication

Multiplication is a fundamental arithmetic operation that involves calculating the total of one number taken times another. In expressions involving multiplication, especially with results from simplifying parentheses, it's crucial to carefully apply multiplication to ensure the correct product.

• Multiply numbers directly
• Watch out for negatives (e.g., negative times positive results in negative)
• Handle integers and decimals with attention

In our exercise, we multiplied the results from the simplified parentheses by numbers outside the parentheses (e.g., calculating \(7 \times -0.9\) and \(-6 \times -4.3\)). Paying close attention to signs is essential since multiplying negative numbers can often trip up calculations.

Subtraction

Subtraction involves taking one number away from another. It is the inverse of addition and can sometimes seem straightforward, but careful attention is required, especially with negative numbers. Fundamental points about subtraction include:

• Order matters (e.g., \(a - b\) is not the same as \(b - a\))
• Subtracting less from more yields positive results; more from less yields negative numbers
• Negative numbers can flip signs (subtracting a negative is adding)

In our exercise, subtraction is the final step, executed between results of the previous multiplication (\(-6.3 - 25.8\)). Handling signs carefully here is crucial as subtracting negative numbers involves adding their positive counterparts. This allows us to smoothly transition from calculation to final, simplified answers.

Negative Numbers

Understanding negative numbers is vital in algebra and arithmetic. They represent values less than zero and have unique properties, particularly when combined with other operations:

• Negative plus negative retains negativity
• Negative times negative becomes positive
• Subtracting a negative equates to adding

In the original exercise, negative numbers appear in parentheses calculations and impact the results (e.g., \(-1.4 - 2.9 = -4.3\)). Approaching negative numbers with patience and understanding their effects in operations like multiplication and subtraction ensures accurate results. The inclusion of multiple negatives requires us to be vigilant to avoid common pitfalls such as flipping signs improperly or miscalculating results.