Problem 102 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{8(-7)+(-2)(-6)}{(-9)(3)+(-10)(-11)}\)

Step-by-Step Solution

Answer

\(-\frac{44}{83}\)

Step 1: Simplify the Numerator

The numerator of the expression is \(8(-7) + (-2)(-6)\). Start by distributing the numbers: \[8 \times (-7) = -56\]\[(-2) \times (-6) = 12\]Now simplify by adding these results: \[-56 + 12 = -44\]

Step 2: Simplify the Denominator

The denominator of the expression is \((-9)(3) + (-10)(-11)\). Start by performing the multiplications:\[(-9) \times 3 = -27\]\[(-10) \times (-11) = 110\]Now simplify by adding these results:\[-27 + 110 = 83\]

Step 3: Simplify the Full Expression

Now that we have simplified the numerator and denominator, combine them into the fraction:\[\frac{-44}{83}\]Since \(-44\) and \(83\) have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Numerator and Denominator

In algebraic expressions like fractions, the terms "numerator" and "denominator" are crucial. These two components dictate how we handle fractions.

• The **numerator** is the top part of a fraction. It tells us how many parts of the whole are being considered or counted.
• The **denominator** is the bottom part. It shows how many equal parts the whole is divided into.

Think of fractions as slices of pizza. If you have a fraction like \( \frac{3}{8} \), you have 3 slices of an 8-slice pizza.
In the exercise, simplifying the numerator involved calculating \(8(-7) + (-2)(-6)\), where the outcome \(-44\) was the simplified numerator. The denominator came from calculating \((-9)(3) + (-10)(-11)\), with the outcome \(83\) as the simplified denominator.
Understanding numerators and denominators helps in simplifying fractions, performing operations, and understanding the value represented by the fraction.

Multiplication of Integers

Multiplication of integers is a basic yet essential skill in solving algebraic expressions. When multiplying integers, the sign of the result depends on the signs of the integers being multiplied.
Here are key rules to remember:

• Multiplying two positive integers yields a positive result.
• Multiplying two negative integers also yields a positive result.
• Multiplying a positive integer with a negative yields a negative result.

For instance, in the original exercise, you have products like \(8 \times (-7) = -56\) and \((-2) \times (-6) = 12\). The first operation shows a multiplication of a positive and a negative integer, leading to a negative result.
The second operation shows a multiplication of two negative integers, leading to a positive result.
Understanding these rules simplifies the process of evaluating more complex expressions and provides the groundwork for solid arithmetic skills.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form, meaning the numerator and denominator share no common factors other than 1.
Steps to simplify include:

• Performing any necessary operations on the numerator and denominator separately as shown in the exercise.
• Finding the greatest common divisor (GCD) of both numbers if they aren't immediately simple.
• Dividing both the numerator and denominator by the GCD to reduce the fraction.

In the example given, \(\frac{-44}{83}\), since \(-44\) and \(83\) are coprime (having no common divisors besides 1), the fraction is already in its simplest form.
Having a fraction in its simplest form makes it easier to work with or compare with other fractions. This process is fundamental in algebra and applied mathematics, helping students navigate through more complex calculations with confidence.