Problem 28 - Algebra A Combined Function

Question

Simplify each expression. Use the distributive property to remove any parentheses. $$ -4(y+6) $$

Step-by-Step Solution

Answer

\(-4(y+6)\) simplifies to \(-4y - 24\).

Step 1: Identify the expression

The given expression is \(-4(y+6)\). This expression involves a multiplication by \(-4\) and an addition inside the parentheses \((y+6)\).

Step 2: Apply the Distributive Property

Use the distributive property, which states \(a(b+c) = ab + ac\), to remove the parentheses. Here, distribute the \(-4\) to both \(y\) and \(6\).

Step 3: Multiply \( -4 \) by \( y \)

Multiply \(-4\) by \(y\): \[-4 imes y = -4y\].

Step 4: Multiply \( -4 \) by \(6\)

Multiply \(-4\) by \(6\): \[-4 imes 6 = -24\].

Step 5: Combine the results

Add the results from Steps 3 and 4 to get the simplified expression: \(-4y - 24\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations.
In our exercise, \(-4(y+6)\) is the algebraic expression we are working with.
The expression consists of:

• an integer coefficient \(-4\), which multiplies the terms inside the parentheses.
• a variable, \(y\), which represents an unknown value.
• a constant, \(6\), which is added to the variable inside the parentheses.

Algebraic expressions are used to model real-life scenarios where quantities are variable.
They are essential for forming equations and functions. Simplifying these expressions often involves strategies like combining like terms or using properties such as the distributive property to make them easier to work with.

Properties of Operations

Understanding the properties of operations is key to simplifying expressions.
These properties help us manipulate and rearrange terms to find solutions.

• Commutative Property states that the order of addition or multiplication does not affect the result: \(a + b = b + a\) or \(a \times b = b \times a\).
• Associative Property highlights that how numbers are grouped in addition or multiplication doesn't change the result: \((a + b) + c = a + (b + c)\) or \((a \times b) \times c = a \times (b \times c)\).
• Distributive Property is key in our example: \(a(b+c) = ab + ac\).
  It allows multiplication to "distribute" over addition or subtraction.

By applying these properties, we can simplify complex expressions, make calculations easier, and find solutions more efficiently.

Simplification of Expressions

The simplification of expressions is the process of making an expression easier to work with or understand.
In the solution provided, the simplification involved applying the distributive property to eliminate parentheses.
Here's how:

• Distribute \(-4\) across the terms in the parentheses: Multiply \(-4\) by \(y\) and \(6\).
  This gives:\(-4y - 24\).
• By distributing, we removed the parentheses and combined terms into simpler forms:
  This reduces the complexity of the expression.

The goal of simplification is to transform the expression into a more straightforward form that is easier to evaluate or substitute values into.
Simplifying expressions is crucial in mathematics, especially as a preparatory step for solving equations or other algebraic manipulations.