Problem 21 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ (2+u) 6 $$

Step-by-Step Solution

Answer

The expression \( (2+u) 6 \) without parentheses, using the distributive property, is \( 12 + 6u \).

Step 1: Identify the Elements

First, identify the values in the expression that match the distributive property model \(a*(b + c)\). In the given expression \( (2+u) 6 \), \(a\) is 6, \(b\) is 2, and \(c\) is \(u\).

Step 2: Apply the Distributive Property

Next, apply the distributive property by multiplying \(a\) with both \(b\) and \(c\), then add the resulting products together. This can be done as follows: 6*2 + 6*\(u\) which results in 12 + 6u.

Key Concepts

Algebraic Expressions

Algebraic expressions are a foundational element in mathematics, representing numbers, variables, and operations all rolled into one. Think of them as mathematical sentences that use symbols and numbers. For example, in the expression \(2 + u\), 2 is a numerical constant and \(u\) is a variable. Variables play a crucial role because they allow expressions to be more flexible and apply to many situations.

In algebraic expressions, we don't just deal with numbers, but also letters that can stand for unknown values or varying quantities. This makes algebra a powerful tool to solve real-world problems. Understanding algebraic expressions is like unlocking a universal language of math that helps you solve equations, model situations, and perform calculations efficiently.

• Numerical Constants: These are fixed numbers like 2 in \(2 + u\).
• Variables: These are represented by letters, like \(u\), and can vary or represent unknowns.
• Operations: These are actions such as addition, represented by "+" in \(2 + u\).

In the example \((2+u) 6\), the goal is to rewrite the expression without parentheses using the distributive property.

Simplifying Expressions

Simplifying expressions makes them easier to work with and understand. It's like cleaning up your room to find everything quickly. The process involves reducing an expression to its simplest form while maintaining its original value. When we simplify, we aim to combine like terms and perform arithmetic operations to get the neatest form possible.

In our example, the expression \((2+u) 6\) can be simplified by applying the distributive property. This allows us to distribute the multiplication over addition, effectively removing the parentheses and simplifying the expression to \(12 + 6u\). This not only makes it easier to read but can also help in solving further equations.

• Combining Like Terms: Adding or subtracting terms that have the same variables.
• Distributive Property: Allows multiplication to be distributed across terms inside a parenthesis, making it easier to simplify.

Simplifying expressions is a key skill in algebra that helps you to manipulate and solve expressions and equations effectively.

Mathematical Operations

Mathematical operations form the backbone of algebra and include addition, subtraction, multiplication, and division. These operations allow us to manipulate numbers and expressions to find a solution. In the context of the distributive property, which is a key operation here, multiplication is distributed over addition inside parentheses.

For instance, in \((2+u) 6\), the multiplication operation is applied to each term inside the parentheses. This expands or distributes the expression into \(6 \times 2 + 6 \times u\). When each part is calculated, it yields \(12 + 6u\). Learning to seamlessly perform these operations allows you to handle more complex algebraic expressions and solve equations efficiently.

• Addition and Subtraction: Combine numbers or expressions to increase or decrease values.
• Multiplication and Division: Scale values up or down, essential for using the distributive property.
• Distributive Property: A specific use of multiplication to simplify expressions.

Mastery of these operations helps transform and solve various mathematical problems effectively and with confidence.