Problem 34 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ 11(y-4) $$

Step-by-Step Solution

Answer

The expression without parentheses is \(11y - 44\).

Step 1: Identify the Expression

The given expression is \(11(y-4)\). This is where we need to apply the distributive property.

Step 2: Apply the Distributive Property

According to the distributive property, \(a(b+c) = ab + ac\). We can distribute \(11\) to both \(y\) and \(-4\). This gives us \(11 imes y + 11 imes (-4)\).

Step 3: Compute the Multiplications

Calculate \(11 imes y\) which results in \(11y\), and \(11 imes (-4)\) which results in \(-44\).

Step 4: Simplify the Expression

Combine the terms from the previous step to get the final expression: \(11y - 44\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. In the exercise we have the expression \(11(y - 4)\). Here, 11 is a constant, \(y\) is a variable, and \(-4\) is another constant. An algebraic expression with variables allows us to represent numbers even when their values are unknown.

Algebraic expressions are used in various fields to model situations, solve problems, and understand relationships between quantities. By learning to manipulate these expressions, like adding them, multiplying them, or distributing elements within them, we unlock the ability to solve algebraic equations and perform simplifications.

Simplification

Simplification involves reducing complex expressions into simpler or more concise forms without changing their values. In our exercise, after using the distributive property, we achieve the expression \(11y - 44\). A simplification ensures the expression is in its most straightforward form.

To simplify algebraic expressions:

• Combine like terms such as adding or subtracting constants.
• Factor common terms out if possible.
• Remember order of operations (PEMDAS/BODMAS).

In \(11y - 44\), the process particularly involved multiplying 11 with each term inside the parenthesis and then bringing those separate calculations together. When no further operation like combining similar terms is possible, your expression is considered simplified.

Multiplication in Algebra

Multiplication in algebra involves applying multiplication principles to numbers and variables. It is crucial for tasks like distributing terms within parentheses, as demonstrated in our exercise. By understanding these principles, you can manipulate and solve algebraic expressions correctly.

In our example:

• Multiply constants with variables: 11 is multiplied by \(y\) to get \(11y\).
• Multiply constants with constants: 11 is multiplied by \(-4\) to result in \(-44\).

This process is a stepping stone to applying the distributive property, which helps in expanding expressions and solving equations more effectively. Multiplication helps in breaking down complex expressions into simpler additive parts.