Problem 29 - Algebra A Combined Function

Question

Simplify each expression. Use the distributive property to remove any parentheses. $$ -5(2 x-3 y+6) $$

Step-by-Step Solution

Answer

The simplified expression is \(-10x + 15y - 30\).

Step 1: Understanding the Problem

We are given an expression \( -5(2x - 3y + 6) \) that needs to be simplified using the distributive property. This means we will multiply each term inside the parentheses by \(-5\).

Step 2: Applying the Distributive Property - First Term

The distributive property tells us to multiply \(-5\) by each term inside the parentheses, one at a time. Start with the first term: \(-5 \times 2x = -10x\).

Step 3: Applying the Distributive Property - Second Term

Next, apply the distributive property to the second term:\(-5 \times (-3y) = 15y\).Notice that multiplying two negative numbers gives a positive result.

Step 4: Applying the Distributive Property - Third Term

Finally, use the distributive property on the third term:\(-5 \times 6 = -30\).

Step 5: Combining the Results

Combine all the results from the individual distributions: \(-10x + 15y - 30\).This is the simplified expression.

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form, making calculations easier. The ultimate goal is to achieve a simplified version that maintains the same value but is expressed in a more straightforward way.

Here are some common steps for simplifying expressions:

• Apply the distributive property to remove parentheses.
• Combine like terms to reduce the expression further.
• Simplify any operations with numbers.

Each step helps streamline the expression, removing any unnecessary complexity.

This process often involves the distributive property, where you multiply a single term by multiple terms inside parentheses. This method is especially important when dealing with expressions involving variables and numbers.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators. These expressions can vary from simple ones, like \(3x + 2\), to more complex expressions with multiple variables and operations.

Algebraic expressions form the foundation of algebra and are essential for solving problems. Key components include:

• **Terms**: Parts of the expression separated by addition or subtraction. For example, in \(2x + 3y - 5\), \(2x\), \(3y\), and \(-5\) are terms.
• **Coefficients**: Numbers multiplying the variables, such as the \(2\) in \(2x\).
• **Constants**: Stand-alone numbers, like \(-5\) in \(2x + 3y - 5\).
• **Variables**: Symbols representing unknown values, such as \(x\) or \(y\).

Understanding these components allows you to manipulate and simplify the expressions effectively.

Negative Numbers

Negative numbers are numbers less than zero, represented by a minus sign \((-\)). They behave differently than positive numbers, especially during multiplication or division.

When handling expressions, remember a few important rules about negatives:

• When multiplying or dividing two negative numbers, the result is positive. For example, \((-5) \times (-3y) = 15y\).
• When multiplying or dividing a positive number with a negative number, the result is negative.

These rules are crucial when using the distributive property, as each multiplication requires careful attention to the signs to avoid errors.

Combining Like Terms

Combining like terms is a method used to simplify expressions by merging terms that have the same variables raised to the same power. This step is essential after distributing terms to further reduce the expression.

Here's how you combine like terms efficiently:

• **Identify Like Terms**: These are terms with identical variable parts. For example, \(2x\) and \(-10x\) are like terms because they both contain the variable \(x\).
• **Add or Subtract Coefficients**: Once you identify like terms, simply add or subtract the coefficients, while keeping the variable part unchanged. For instance, \(2x + (-10x)\) simplifies to \(-8x\).

This process helps in further simplifying the expression and is a critical skill in algebra for achieving the simplest form of an expression.