Problem 101 - College Algebra

Question

Write each algebraic expression without parentheses. \(\frac{1}{3}(3 x)+[(4 y)+(-4 y)]\)

Step-by-Step Solution

Answer

The algebraic expression simplifies to \(x\).

Step 1: Distribute Multiplication Over Addition/Subtraction

Distribute the \(1/3\) by multiplying it with the \(3x\) which is inside the parentheses. Similarly, the brackets contain two like terms, \(4y\) and \(-4y\), which can be combined together.\nSo, the algebraic expression simplifies to \(1/3 * 3x + 4y - 4y\).

Step 2: Perform the Multiplication

Multiply \(1/3\) and \(3x\) together which results in \(x\).\nThe algebraic expression further simplifies to \(x + 4y - 4y\).

Step 3: Combine Like Terms

Add together the like terms \(4y\) and \(-4y\). This results in \(0y\) which equals to \(0\).\nTherefore, the algebraic expression further simplifies to \(x + 0\).

Step 4: Final Simplification

The term adds \(0\) to \(x\) which doesn’t change the value of \(x\).\nTherefore, the final algebraic expression without parentheses is \(x\).

Key Concepts

Distributive Property

The distributive property is a fundamental aspect of algebra that allows you to simplify expressions by "distributing" multiplication over addition or subtraction. This means you can multiply a term outside the parentheses by each term inside the parentheses separately.
For example, consider the expression \(\frac{1}{3}(3x)\). By applying the distributive property, you multiply \ \frac{1}{3} \ by \3x\, resulting in \ x \ because \(\frac{1}{3} imes 3x = x\). This property helps in breaking down complex expressions into simpler terms, making it easier to work with them in further algebraic processes.

Like Terms

In algebra, like terms refer to terms that have the exact same variable part and can thus be added or subtracted. When expressions are combined, identifying like terms helps streamline the process.
For instance, in the expression \(4y + (-4y)\), \4y\ and \ -4y\ are like terms because they have the same variable, \ y\. You can therefore combine them to simplify the expression: \4y - 4y = 0\. This step is important as it reduces the complexity of the expression, leading us closer to the simplest form.

Simplification

Simplification involves reducing an algebraic expression to its simplest form by performing operations like distributing, combining like terms, and removing unnecessary elements. The goal is to have an expression that is easy to understand and evaluate.
Take the expression \ x + 4y - 4y \. After identifying the like terms \( 4y \ and \ -4y \) and combining them to get zero, you simplify the expression to \ x + 0 \, which further simplifies to \ x \. Each simplification step eliminates redundancies, helping you to focus on the core components of the expression.

Parentheses Removal

Removing parentheses is a straightforward task in algebra typically achieved through the application of the distributive property or by simplifying the terms inside them. This step is crucial because it transforms a complex expression into a more manageable one.
Initially, the expression was \( \frac{1}{3}(3x) + [(4y)+(-4y)] \). As you used the distributive property and combined like terms, the parentheses became unnecessary, resulting in \ x + 0 \. This final step of completely removing the parentheses leaves you with the simplified expression \ x \, ensuring clarity and ease of calculation in further problems.