Problem 31 - Algebra A Combined Function

Question

Simplify each expression. Use the distributive property to remove any parentheses. $$ -(3 x-2 y+1) $$

Step-by-Step Solution

Answer

The simplified expression is \(-3x + 2y - 1\).

Step 1: Apply the Distributive Property

We start by applying the distributive property to simplify the expression. The expression given is \[-(3x - 2y + 1)\]. This can be rewritten by distributing the negative sign to each term inside the parentheses, effectively multiplying each term by -1.

Step 2: Distribute the Negative Sign

By applying the distributive property, we get:\[-1 \cdot 3x - 1 \cdot (-2y) - 1 \cdot 1\].Now let's perform the multiplication on each term: - The first term is \[-1 \cdot 3x = -3x\].- The second term is \[-1 \cdot (-2y) = 2y\] (note the double negative becomes positive).- The third term is \[-1 \cdot 1 = -1\].

Step 3: Combine All Results

After multiplying, combine all the results from step 2 into a single expression:\[-3x + 2y - 1\].This is the simplified form of the original expression after applying the distributive property.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental aspect of algebra that helps make complex calculations more manageable. At its core, simplification involves reducing an expression to its simplest form. This can involve several techniques, such as:

• Combining like terms: These are terms that have identical variable parts. For instance, in the expression \(3x + 2x\), the terms can be combined to \(5x\).
• Applying the distributive property: This is particularly useful when dealing with expressions involving parentheses. For example, distributing a factor across terms in parentheses helps eliminate parentheses, making the expression easier to work with.

The goal of simplification is to make expressions cleaner and often shorter, enabling easier computation or further algebraic manipulation. Simplification is crucial for solving equations, as it often reveals relationships between variables.

Multiplying Integers

Multiplying integers is a basic arithmetic operation, crucial in algebra for manipulating expressions. The rules for multiplying integers focus on understanding how the product's sign is determined:

• Positive \(\times\) Positive = Positive: For example, \(3 \times 4 = 12\).
• Negative \(\times\) Negative = Positive: Multiplying two negative numbers, such as \((-3) \times (-4) = 12\), results in a positive product.
• Positive \(\times\) Negative = Negative: This occurs when a positive number is multiplied by a negative number, e.g., \(3 \times (-4) = -12\).
• Negative \(\times\) Positive = Negative: Similarly, multiplying a negative number by a positive number leads to a negative result, such as \((-3) \times 4 = -12\).

Understanding these rules is essential when simplifying expressions, especially during the application of the distributive property, as illustrated in our exercise. Selecting and consistently applying these rules ensures accuracy in algebraic manipulations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that form the building blocks of algebra. They can range from simple expressions like \(x + 2\) to more complex forms involving multiple terms and operations, such as \((-3x + 2y - 1)\). Here are some key features:

• Variables: Represent unknowns and are usually denoted by symbols such as \(x\), \(y\), or \(z\).
• Constants: Known values, expressed as numbers, like "3" in \(3x\).
• Coefficients: Numbers that multiply the variables, such as "3" in \(3x\). They indicate how many times the variable is counted.
• Operations: Include addition, subtraction, multiplication, and division, which combine the constants and variables.

When working with algebraic expressions, understanding how to rearrange and simplify them using properties such as the distributive property is vital. These expressions serve as the foundation for forming equations and solving problems in algebra.