Problem 90 - Intermediate Algebra

Question

Simplify. See Example \(6 .\) $$10\left[\frac{3}{5}(2 s+2 t)-\frac{4}{5}(s-t)+1\right]$$

Step-by-Step Solution

Answer

The simplified expression is \( 4s + 20t + 10 \).

Step 1: Distribute Inside the Brackets

Begin by distributing the fractions inside the brackets. For \( \frac{3}{5}(2s + 2t) \), distribute \( \frac{3}{5} \) across \( 2s \) and \( 2t \). Similarly, for \( \frac{4}{5}(s - t) \), distribute \( \frac{4}{5} \) across \( s \) and \( -t \). This becomes:\[ \frac{3}{5}(2s) + \frac{3}{5}(2t) = \frac{6}{5}s + \frac{6}{5}t \]\[ \frac{4}{5}(s) - \frac{4}{5}(t) = \frac{4}{5}s - \frac{4}{5}t \]

Step 2: Substitute and Simplify Inside the Brackets

Replace the terms in the original expression with the simplified expressions found in Step 1:\[ 10 \left[ \frac{6}{5}s + \frac{6}{5}t - \left(\frac{4}{5}s - \frac{4}{5}t\right) + 1 \right] \]Now, combine like terms:\[ \frac{6}{5}s - \frac{4}{5}s = \frac{2}{5}s \]\[ \frac{6}{5}t + \frac{4}{5}t = \frac{10}{5}t = 2t \]

Step 3: Combine and Simplify Further

Substitute the combined terms back into the brackets:\[ 10 \left[ \frac{2}{5}s + 2t + 1 \right] \]Combine the fraction and whole numbers; remember here that adding \( 1 \) is like adding \( \frac{5}{5} \). Thus, the expression becomes:\[ 10 \left[ \frac{2}{5}s + 2t + \frac{5}{5} \right] \]

Step 4: Distribute the 10 Across the Expression

Multiply each term inside the brackets by 10:\[ 10 \times \frac{2}{5}s = 4s \]\[ 10 \times 2t = 20t \]\[ 10 \times 1 = 10 \]Putting it all together gives:\[ 4s + 20t + 10 \]

Key Concepts

Simplification

Simplification in algebra involves expressing an expression in its simplest form. This means reducing the complexity of expressions while maintaining equality with the original form. Simplifying expressions makes them easier to work with, especially for further calculations or comparisons. In our exercise, simplifying involves tackling several smaller steps, such as applying the distributive property and then combining like terms. These steps break down complex expressions into manageable pieces. Thus, understanding simplification ensures that you can solve algebraic problems in a structured way, ultimately leading to a clean and neat final answer.

Distributive Property

The distributive property is a vital algebra concept used to simplify expressions. It helps to eliminate brackets by distributing a factor over terms inside the bracket. For the expression in our exercise, the distributive property is applied twice. First, we distribute the fraction \( \frac{3}{5} \) across \( (2s + 2t) \).

• This results in \( \frac{3}{5} \times 2s + \frac{3}{5} \times 2t = \frac{6}{5}s + \frac{6}{5}t \).

Similarly, we distribute \( \frac{4}{5} \) over \( (s - t) \), giving:

• \( \frac{4}{5} \times s - \frac{4}{5} \times t = \frac{4}{5}s - \frac{4}{5}t \).

This step breaks down complicated expressions into simpler terms, facilitating the combination of like terms next.

Combining Like Terms

Combining like terms means grouping and simplifying terms that have identical variables. This step is essential for the expression's final simplification. Like terms are added or subtracted by managing their coefficients. Let's look at the terms in our exercise:

• First, for \( s \) terms: \( \frac{6}{5}s - \frac{4}{5}s = \frac{2}{5}s \).
• Second, for \( t \) terms: \( \frac{6}{5}t + \frac{4}{5}t = \frac{10}{5}t = 2t \).

By performing these calculations, we consolidate the variable terms, leading to a compact and manageable expression. Subsequently, this sets the stage for multiplying by 10 and arriving at the simplified whole expression. Combining like terms simplifies algebraic expressions dramatically, making further calculations quicker and more accurate.