Problem 77 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -\frac{1}{3}(9 x-4) $$

Step-by-Step Solution

Answer

-3x + \frac{4}{3}

Step 1: Understand the Distributive Property

The distributive property states that for any numbers or expressions a, b, and c, the following holds: \[ a(b + c) = ab + ac \] and similarly for subtraction: \[ a(b - c) = ab - ac \].

Step 2: Identify the Terms in the Expression

In the expression \[ -\frac{1}{3}(9x - 4), \] the scalar (factor outside the parentheses) is \[ -\frac{1}{3}, \] and the terms inside the parentheses are \[ 9x \] and \[ -4. \]

Step 3: Apply the Distributive Property

Distribute \[ -\frac{1}{3} \] to each term inside the parentheses: \[ -\frac{1}{3} \times 9x \] and \[ -\frac{1}{3} \times (-4). \]

Step 4: Simplify Each Product

Calculate each product separately: \[ -\frac{1}{3} \times 9x = -3x \] and \[ -\frac{1}{3} \times (-4) = \frac{4}{3}. \]

Step 5: Combine the Results

Combine the simplified terms to rewrite the original expression: \[ -3x + \frac{4}{3}. \]

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra. It helps in simplifying and solving expressions. This property shows how multiplication interacts with addition and subtraction within parentheses. Remember, if you have an expression like \[a(b + c)\], it becomes \[ab + ac\]. Similarly, \[a(b - c)\] becomes \[ab - ac\].

For example, consider the expression from the exercise: \[-\frac{1}{3}(9x - 4)\]. Here, the distributive property tells us to multiply \(-\frac{1}{3}\) with both \(9x\) and \(-4\). This step-by-step method reveals how each part of the expression is handled independently before recombining.

Simplifying Expressions

Simplifying expressions is about making them more manageable or solving them. By using properties like the distributive property, we break complex expressions into simpler ones. Taking our exercise example:

• First, we identify the terms inside the parentheses: \(9x\) and \(-4\).
• Then, we apply the scalar outside the parentheses (\(-\frac{1}{3}\)) to each term individually.

Once these parts are independently simplified, we can recombine them to get a clean, straightforward expression.

For instance, in this exercise, multiplying \(-\frac{1}{3} \times 9x\) gives \(-3x\) and multiplying \(-\frac{1}{3} \times -4\) gives \(\frac{4}{3}\). Our simplified expression then becomes \(-3x + \frac{4}{3}\).

Algebraic Multiplication

Algebraic multiplication involves multiplying coefficients and variables. When working with algebraic expressions, coefficients (numbers) and variables (letters) should be treated carefully.

Using our exercise example again:

• First, focus on multiplying the coefficients: \(-\frac{1}{3}\) and \(9\), which simplifies to \(-3\).
• Next, apply this result to the variable \(x\), giving us \(-3x\).
• Then, multiply the scalar \(-\frac{1}{3}\) with the constant \(-4\), resulting in \(\frac{4}{3}\).

Putting it all together, the algebraic multiplication step helps break down the expression into more manageable pieces, eventually leading to the solution \(-3x + \frac{4}{3}\).

Each step ensures that we're methodically applying algebraic multiplication to both terms inside the parentheses, following the distributive property rule.