Problem 350 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 10\left(\frac{3}{10} x-\frac{2}{5}\right) $$

Step-by-Step Solution

Answer

3x - 4

Step 1: Identify the terms inside the parentheses

The expression inside the parentheses is \( \frac{3}{10} x - \frac{2}{5} \).

Step 2: Apply the Distributive Property

According to the Distributive Property, \( a(b + c) = ab + ac \). Apply this property by multiplying 10 with each term inside the parentheses.

Step 3: Multiply each term by 10

Multiply 10 by \( \frac{3}{10} x \) and \( \frac{2}{5} \): \[ 10 \left( \frac{3}{10} x \right) - 10 \left( \frac{2}{5} \right) \]

Step 4: Simplify each multiplication

Calculate the products: \( 10 \left( \frac{3}{10} x \right) = 3x \) and \( 10 \left( \frac{2}{5} \right) = 4 \)

Step 5: Write the final simplified expression

The final simplified expression is: \[ 3x - 4 \]

Key Concepts

simplify algebraic expressions

Simplifying algebraic expressions means rewriting them in a simpler form. This is achieved by combining like terms and applying mathematical properties, such as the Distributive Property. In the example, we start with the expression \( 10 \bigg( \frac{3}{10} x - \frac{2}{5} \bigg) \). The goal is to make it easier by performing operations that eliminate parentheses.

Firstly, identify the terms inside the parentheses: \( \frac{3}{10} x \) and \( - \frac{2}{5} \). Combining these terms wouldn't be practical, so we use the Distributive Property. This property helps in breaking down complex expressions into simpler parts by distributing the outside term to each inside term separately.

Hence, we apply the Distributive Property by multiplying \( 10 \) with each term inside the parentheses:

• \( 10 \times \frac{3}{10} x \rightarrow 3x \)
• \( 10 \times - \frac{2}{5} \rightarrow -4 \)

After simplifying the multiplication, we get the expression \( 3x - 4 \). This is simpler and easier to understand compared to the original, more complex form.

multiplication of fractions

Understanding the multiplication of fractions is crucial for simplifying algebraic expressions and solving equations. Let's break it down using the example from our exercise.

We start with fraction multiplication in the expression \( 10 \bigg( \frac{3}{10} x - \frac{2}{5} \bigg) \). Specifically, we need to multiply the whole number 10 by each fraction. Here's how:

• \( 10 \times \frac{3}{10} x \) - In this multiplication, the 10s in the numerator and denominator cancel each other out, simplifying the expression to \( 3x \).
• \( 10 \times - \frac{2}{5} \) - To multiply a whole number by fraction, you multiply the numerator by the whole number while keeping the denominator the same. Here, it's \( 10 \times \frac{2}{5} = \frac{20}{5} \), which simplifies to 4.

Remember that when multiplying fractions:

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify the fraction if possible.

For the expression we are working with, the multiplication steps result in \( 3x \) and \( -4 \). This shows us that working with fractions can simplify complex algebraic problems.

linear equations

Linear equations form the backbone of algebra and problem-solving. They are equations of the first degree, meaning the highest power of the variable is 1. Simplified linear expressions, like \( 3x - 4 \), are straightforward to solve and understand.

The original problem involved simplifying an expression to this form. Why do we value such forms? Because they lead directly to solving equations like \(3x - 4 = 0\), where finding the value of x is simple. Here's a quick overview of the steps involved:

• Simplify both sides of the equation, if necessary.
• Isolate the variable term on one side, often using addition or subtraction.
• Divide or multiply to solve for the variable.

Example: Solve for \( x \) in the equation \( 3x - 4 = 0 \).
Step 1: Add 4 to both sides -> \( 3x = 4 \).
Step 2: Divide both sides by 3 -> \( x = \frac{4}{3} \).

This makes it clear how simplified algebraic expressions can lead to solving linear equations efficiently. These concepts are closely linked, underscoring the importance of mastering each step.