Problem 47 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{-44-8 t}{-4} $$

Step-by-Step Solution

Answer

The simplification of the expression \( \frac{-44-8t}{-4} \) is 11 + 2t.

Step 1: Identify and Write Down the Expression

First, write down the expression which is \( \frac{-44-8 t}{-4} \).

Step 2: Distribute the Denominator

Next, distribute the denominator over the terms in the numerator such that the expression becomes \( -44/-4 + -8t/-4 \).

Step 3: Simplify Each Term

Then, simplify each term in the resulting expression from step 2. So, \( -44/-4 \) simplifies to 11 and \( -8t/-4 \) simplifies to 2t.

Step 4: Write the Final Simplified Expression

Finally, the simplified expression is written as 11+2t.

Key Concepts

Distributive Property

Understanding the distributive property is crucial when working with algebraic expressions. It's a rule that allows you to multiply a number by a sum or difference inside parentheses. In simpler terms, this operation 'distributes' the number outside of the parentheses to each term inside. The formula for the distributive property is: \( a(b+c) = ab + ac \).

In the given exercise, we apply the distributive property in reverse. Instead of distributing multiplication, we're distributing division across the sum \( -44-8t \). This means we divide both terms in the numerator separately by the denominator \( -4 \), resulting in \( -44/-4 \ and \ -8t/-4 \). It's like sharing a pie equally among friends; every term gets its fair share of division!

Simplifying Fractions in Algebra

When it comes to simplifying fractions in algebra, the goal is to make the expression as straightforward as possible. Simplifying is like decluttering your workspace; it's all about removing the unnecessary to focus on what's truly important. With numbers, it involves dividing both the numerator and the denominator by their greatest common factor until you can't simplify any further.

In our exercise, once we've distributed the denominator, we're left with fractions \( -44/-4 \) and \( -8t/-4 \) to simplify. Since \( -44 \) and \( -4 \) are both divisible by \( -4 \), the result for the first term is 11. Similarly, \( -8t \) and \( -4 \) can be reduced to 2t. Remember, simplifying algebraic fractions also includes reducing literal coefficients (those with variables), just as with numerical coefficients.

Combining Like Terms

The concept of combining like terms is a bit like organizing a set of tools: you group similar items together to create order and simplify your workspace. In algebra, 'like terms' are terms that have the same variable raised to the same power—think of them as a matching set. When combining them, you add or subtract the coefficients and keep the variable part unchanged.

In the context of our exercise, after simplifying each fraction, we're left with distinct terms: 11 and 2t. Since they're not like terms—one is a constant and the other is a variable term—they can't be combined further. But the understanding of how to identify and combine like terms is essential when dealing with more complex expressions where such terms are present. Keeping like terms together makes the algebraic expressions neat and often reveals patterns that can be crucial for solving equations.