Problem 37 - Prealgebra

Question

Apply the distributive property to each expression and then simplify. $$8(2 y+4)+3 y$$

Step-by-Step Solution

Answer

The simplified expression is \(19y + 32\).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). Here, apply the distributive property to the term \(8(2y+4)\). This gives us:\[8 \cdot 2y + 8 \cdot 4\]Calculate these products to get:\[16y + 32\]

Step 2: Combine Like Terms

Now simplify the expression by combining like terms. The original expression is:\[16y + 32 + 3y\]Combine the \(y\) terms \(16y\) and \(3y\):\[(16y + 3y) + 32 = 19y + 32\]

Step 3: Final Simplification

Ensure all like terms are combined and the expression is fully simplified. The final simplified expression is:\[19y + 32\]

Key Concepts

Simplifying Expressions

Simplifying expressions is a key concept in algebra that involves reducing expressions to their simplest form. This makes them easier to work with and understand. In our exercise, we have the expression \(8(2y + 4) + 3y\), which needs to be simplified.

• First, you apply mathematical operations such as the distributive property to break down complex expressions.
• This involves distributing multipliers across terms within parentheses.
• Once expanded, you will need to look out for like terms to combine.

Utilizing the distributive property, we transformed \(8(2y + 4)\) into \(16y + 32\). This is one crucial step in simplifying. From here, combining like terms further reduces the expression, eventually arriving at the simplest form: \(19y + 32\). The essence of simplification is to achieve an expression that is easy to read and use in further calculations.

Combining Like Terms

Combining like terms is another vital skill in algebra, especially when simplifying expressions. A 'like term' in algebra refers to terms that have identical variables raised to the same powers, which makes them compatible for addition or subtraction.

• In our expression \(16y + 32 + 3y\), the terms \(16y\) and \(3y\) are 'like terms' because they both contain the variable \(y\).
• Adding these like terms together is simple: just add their coefficients. Here, \(16 + 3 = 19\), resulting in \(19y\).

The step of combining like terms allows for a streamlined expression by unifying similar components, effectively reducing the number of terms to look at, and simplifying calculations.

Algebraic Expressions

Algebraic expressions form the foundation of algebra and are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

• An algebraic expression does not have an equal sign, which distinguishes it from an equation.
• The exercise expression, \(8(2y + 4) + 3y\), is a typical example, consisting of terms like \(8(2y + 4)\) and \(3y\).
• Transforms, such as distributing and combining like terms, enable simplification of these expressions.

The manipulation and simplification processes of algebraic expressions are crucial for solving equations, understanding functions, and handling mathematical models. Mastery in managing these elements forms the core of algebra skills, facilitating a deeper comprehension of mathematics as a whole.