Problem 25 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 3(2 x-5)-4(5 x-2) $$

Step-by-Step Solution

Answer

The simplified expression is \(-14x - 7\).

Step 1: Distribute Constants

Start by applying the distributive property to each term that is in parentheses. Multiply each term inside the parenthesis by the number outside. For the first part, distribute 3 across the terms inside the first set of parentheses: \[ 3(2x - 5) = 6x - 15 \]For the second expression, distribute -4 across the terms inside the parenthesis:\[ -4(5x - 2) = -20x + 8 \]

Step 2: Combine Like Terms

Now that the expression has no parentheses, combine like terms. You have: \[ 6x - 15 - 20x + 8 \]First, combine the like terms involving \(x\):\[ 6x - 20x = -14x \]Then, combine the constant terms:\[ -15 + 8 = -7 \]

Step 3: Write the Final Expression

Now combine the results from combining the like terms in step 2:\[-14x - 7\]This is the simplified version of the original expression.

Key Concepts

Distributive Property

In algebra, one of the foundational concepts is the distributive property. It allows us to multiply a single term across the terms inside a set of parentheses. This means you distribute or "pass out" the multiplication across each term within the parentheses. When faced with the expression \(3(2x - 5) - 4(5x - 2)\), we need to multiply 3 by both \(2x\) and \(-5\), and similarly, multiply \(-4\) by \(5x\) and \(-2\).
Below is a detailed breakdown:

• First, distribute the 3: \(3 \cdot 2x = 6x\) and \(3 \cdot (-5) = -15\), resulting in \(6x - 15\).
• Next, distribute \(-4\): \(-4 \cdot 5x = -20x\) and \(-4 \cdot (-2) = 8\), resulting in \(-20x + 8\).

This step is vital because it transforms expressions with parentheses into simpler expressions that are easier to combine. The distributive property thus serves as a powerful tool to simplify and solve equations effectively.

Combining Like Terms

Once the distributive property has been applied and the parentheses are removed, the next step is to combine like terms. This is critical for simplification because it consolidates similar elements of the expression. Like terms are terms that contain the same variable raised to the same power. In the expression \(6x - 15 - 20x + 8\):

• "Like terms" involving \(x\) can be combined: \(6x - 20x = -14x\).
• The constant terms are also combined: \(-15 + 8 = -7\).

Always remember, terms without variables are combined separately from those with variables. By systematically combining like terms, the expression becomes more succinct. This method of rearrangement and consolidation is pivotal for solving algebraic problems seamlessly.

Simplification of Expressions

After applying the distributive property and combining like terms, you're left with the most simplified form of the algebraic expression. Simplification is about reducing the expression to its most compact form without changing its value. In our example, after performing those steps, the expression is simplified to \(-14x - 7\).

• No further simplification is required as each term within the expression is its simplest form.
• It's crucial to always check that all like terms have been combined and ensure no further common factors are present.
  

Being able to recognize when an expression is fully simplified is just as important. A simplified expression is easier to interpret and solve in further applications, whether in solving equations or modeling real-world scenarios.