Problem 98 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. Simplify the expression in each exercise. $$ \frac{2(-5)-1(-4)}{5(-5)-6(-4)} $$

Step-by-Step Solution

Answer

The simplified expression for the given problem is 6.

Step 1: Process Numerator

In the numerator, you have two operations, multiplication and subtraction. Multiply each of the respective numbers together using their signs. So, it would be \(2 \times -5\) which equals -10, and \(1 \times -4\) which equals -4.

Step 2: Process Numerator Subtraction

Subtract the results in the numerator. It would be \(-10 - (-4)\), keeping in mind that subtracting a negative is similar to adding a positive. Hence, the simplified numerator becomes \(-10 + 4\) which equals -6.

Step 3: Process Denominator

In the denominator, you again have two operations, multiplication and subtraction. Carry out these multiplication operations like in the numerator to get \(5 \times -5\) which equals -25, and \(-6 \times -4\) which equals 24.

Step 4: Process Denominator Subtraction

Subtract the results in the denominator. It would be \(-25 - (-24)\), similar to addition, so the simplified denominator becomes \(-25 + 24\) which equals -1.

Step 5: Simplify Fraction

Finally, simplify the fraction using the results from the numerator and the denominator. The fraction \(-6 / -1\) simplifies to 6 as a negative divided by a negative gives a positive result.

Key Concepts

Fraction Simplification

Simplifying algebraic fractions is very much akin to reducing numerical fractions to their simplest form, where we aim to break down the expression so it cannot be made any simpler using basic arithmetic operations.
Start by looking at the individual components of the fraction—namely, the numerator (top part) and the denominator (bottom part). Each must be simplified individually before assessing the fraction as a whole. Make sure to apply the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), as you work through each component.

• Process the numerator and denominator separately.
• Always perform multiplication and division before addressing addition and subtraction.
• Look for opportunities to combine like terms or apply the distributive property.
• Finally, divide the simplified numerator by the simplified denominator to get your simplest form.

Following these guidelines will yield a neat and tidy result, free from unnecessary complexity.

Numerical Operations

Mastering numerical operations is essential when working with algebraic expressions, especially in fraction simplification. In our exercise, multiplication and subtraction are the two operations we need to perform first within both the numerator and the denominator.
Remember that multiplication involving negative numbers follows these rules: a negative times a positive gives a negative result, and a negative times a negative gives a positive result.

• When multiplying, consider the signs of the numbers involved.
• For subtraction, recognize that subtracting a negative number is the same as adding its positive counterpart.
• Always double-check your calculations to prevent sign errors, which are common mishaps.

It's helpful to write down intermediate steps to keep track of your work, ensuring that no detail is overlooked in the simplification process. Understanding these operations intimately paves the way for smooth manipulation of more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions represent a cornerstone concept in algebra, encapsulating numbers, variables, and arithmetic operations to describe patterns, relationships, and general mathematical statements. Simplification of such expressions, as shown in the above exercise, is a fundamental skill that involves reducing the expression to its most compact, efficient form.

• Identify and perform operations within parentheses first.
• Use the associative and commutative properties of multiplication to rearrange and combine like terms.
• Apply inverse operations to isolate terms and simplify the expression.

By understanding how to handle the numerical parts of an expression, you can more easily generalize these skills to work with algebraic terms, including those with variables. As these components are simplified, the overall expression clarifies, ultimately leading to increased comprehension and the ability to solve complex problems with confidence.