Problem 101 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{-7(-1)+(-3) 4}{(-2)(5)+(-6)(-8)}\)

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{-5}{38}\).

Step 1: Evaluate the Numerator Operations

First, evaluate the operations in the numerator of the fraction:\[-7(-1) + (-3)(4)\]Multiply \[-7 \times -1 = 7\] and\[-3 \times 4 = -12\].Then, sum the results: \[7 + (-12) = 7 - 12 = -5\].The numerator is \(-5\).

Step 2: Evaluate the Denominator Operations

Now, evaluate the operations in the denominator:\[(-2)(5) + (-6)(-8)\]Multiply \[(-2) \times 5 = -10\] and \[(-6) \times (-8) = 48\].Then, add the results: \[-10 + 48 = 38\].The denominator is \(38\).

Step 3: Formulate the Fraction

Construct the fraction using the results from Steps 1 and 2. The fraction becomes:\[\frac{-5}{38}\]

Step 4: Simplify the Fraction

Check if the fraction \(\frac{-5}{38}\) can be simplified. Since there are no common factors between 5 and 38 other than 1, the fraction is already in its simplest form.

Key Concepts

Fraction Evaluation

Understanding how to evaluate fractions involves handling both the numerator and the denominator separately before forming the complete fraction. The fraction is divided into two major parts:

• The numerator - this part is on the top of the fraction.
• The denominator - this part is on the bottom of the fraction.

To evaluate a fraction, you need to solve any arithmetic operations in both these parts separately. Always remember: perform multiplication and division first, following the order of operations, before addition and subtraction. Evaluating fractions accurately is crucial when the expressions in the numerator and denominator are more complex.
Take each step at a time, and don't rush the calculations. Breaking down fraction evaluation into smaller steps helps in avoiding mistakes. This is particularly useful when you encounter negative numbers or multiple terms in the expression.

Numerator and Denominator

Every fraction consists of two parts: the numerator and the denominator. Knowing their roles will help you understand fraction evaluation better. The numerator acts as the "top" number of a fraction. It represents how many parts you have. For example, if you have a pizza cut into 8 slices and you take 3, the numerator will be 3. The denominator, on the other hand, is the "bottom" number and tells you into how many parts the whole is divided. Following the pizza example, the denominator is 8 since the pizza was divided into 8 slices. In mathematical operations, treat each part separately:

• First, solve and simplify any operations in the numerator.
• Then, handle the operations in the denominator.

Construct the whole fraction by combining the evaluated numerator and denominator. This approach will lead you to the correct evaluation of the fraction. Always double-check the arithmetic from each part for accuracy.

Multiplication with Negative Numbers

Multiplying with negative numbers can sometimes be tricky but is crucial for handling algebraic expressions accurately. When you multiply two negative numbers, the product is positive. For instance, egin{align*} (-2) imes (-3) = 6 demonstrates how the negatives cancel each other out. However, if you multiply a negative number by a positive number, the result is negative. For example, egin{align*} (-2) imes 3 = -6. Here's a simple rule to remember:

• Negative times negative gives a positive.
• Negative times positive yields a negative.

Keeping these rules in mind helps avoid common arithmetic errors, especially in complex algebraic expressions where negative numbers are involved. These principles are fundamental parts of simplifying and evaluating algebraic expressions accurately.

Simplifying Fractions

To simplify a fraction means to reduce it to its simplest form. A fraction is simplified if the numerator and denominator have no common factors other than 1. The fraction \(\frac{-5}{38}\) is already simplified.To simplify fractions generally, follow these steps:

• Identify the greatest common factor (GCF) of both the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.
• Check if the fraction can be simplified further. Repeat the process if necessary.

For instances where no common factors exist except for 1, the fraction remains as it is. Simplifying fractions is a useful skill to aid in both clarity and accuracy, especially when comparing different fractions or preparing to add and subtract them. Simplification helps in cutting down on complicated answers and makes it easier to work with fractions in subsequent operations.