Problem 97 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{6-2(-3)}{4-3(-2)}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{6}{5}\).

Step 1: Distribute the negative sign

First, distribute the negative signs inside the parentheses in both the numerator and the denominator of the fraction. For the numerator, we have \(-2(-3)\). Distributing the negative sign gives us \(+6\). So, the numerator is now \(6 + 6\). For the denominator, \(-3(-2)\) becomes \(+6\), turning it into \(4 + 6\).

Step 2: Simplify the expressions

Now, simplify the expressions obtained in Step 1. The numerator becomes \(6 + 6 = 12\). The denominator becomes \(4 + 6 = 10\).

Step 3: Divide the simplified values

Finally, divide the simplified numerator by the simplified denominator to solve the expression: \(\frac{12}{10}\).

Step 4: Simplify the fraction

Simplify the fraction \(\frac{12}{10}\) by dividing both the numerator and the denominator by their greatest common divisor, which is \(2\). This gives \(\frac{12 \div 2}{10 \div 2} = \frac{6}{5}\).

Key Concepts

Distributive Property

When faced with expressions that contain operations inside parentheses, the distributive property is a powerful tool to simplify them. This principle states that multiplying a sum or difference by a number is the same as doing each multiplication separately. For instance, the distributive property for

• numerator as \( -2(-3) \) and
• denominator as \( -3(-2) \)

converts them into \( +6 \) because a negative multiplied by a negative results in a positive.
This is crucial when simplifying complex expressions, as it allows the elimination of parentheses and simplifies the calculation process. Here, applying the distributive property helped us to transform the expression inside the parentheses to another, easier-to-use form, ultimately simplifying the algebraic expression greatly.

Simplifying Fractions

Once you've worked through an expression and have a fraction to work with, simplification is key to getting the simplest form of your answer. Simplifying fractions means dividing both the numerator and the denominator by their greatest common divisor (GCD).
In our exercise, we reached the fraction \( \frac{12}{10} \). The GCD of 12 and 10 is 2. So, we divide the numerator and the denominator by 2 to simplify this fraction, resulting in \( \frac{6}{5} \).

• This step ensures that your answer is in its simplest form.
• Simplified fractions are crucial in algebra because they are easier to work with and understand.

Always remember, finding the GCD can sometimes involve prime factorization or just recognizing common factors for speed and efficiency.

Order of Operations

Solving any algebraic expression requires a strict adherence to the order of operations, often recalled by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In this exercise, after using the distributive property, we continued to use the order of operations to solve the expression inside the fraction. First, the operations within the parentheses were resolved. Following that, the results were simplified. This orderly process is essential not just for getting the correct answer, but also for understanding the flow and structure of solving equations in algebra.
Adhering to the order of operations is fundamental in avoiding common pitfalls and errors in solutions, ensuring clarity and accuracy.