Problem 356 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ u(v-10) $$

Step-by-Step Solution

Answer

\( uv - 10u \)

Step 1: Understand the Distributive Property

The Distributive Property states that for any numbers or variables, a, b, and c, the following is true: \[ a(b + c) = ab + ac \] We can also apply it when you have a subtraction inside the parentheses: \[ a(b - c) = ab - ac \]

Step 2: Identify the Terms

Identify the terms and factors in the expression \( u(v - 10) \): - The term outside the parentheses is \( u \). - The terms inside the parentheses are \( v \) and \( -10 \).

Step 3: Distribute the Multiplication

Apply the Distributive Property by multiplying \( u \) with each term inside the parentheses: \[ u(v - 10) = u \times v - u \times 10 \]

Step 4: Simplify the Expression

Simplify the expression by performing the multiplication: \[ u \times v - u \times 10 = uv - 10u \]

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing an algebraic expression to its simplest form, making it easier to work with. By simplifying, we combine like terms, eliminate unnecessary parentheses, and perform arithmetic operations where possible.
In the exercise given, we started with the expression \(u(v - 10)\). By using algebraic rules, we broke it down into smaller parts, making it simpler and easier to understand. This process helps in solving equations and understanding relationships between variables.
Understanding how to simplify expressions is crucial for tackling more complex algebraic problems in the future.

Algebraic Multiplication

Algebraic multiplication is the process of distributing and combining variables and numbers. This is a key step in simplifying expressions.
In our example, we multiplied the term outside the parentheses, \(u\), by each term inside the parentheses, \(v - 10\).
Specifically, we applied the multiplication as follows:

• First, multiply \(u\) by \(v\), resulting in \(uv\).
• Next, multiply \(u\) by \(-10\), giving \(-10u\).

Putting it all together, the distributed expression becomes \(uv - 10u\).
This step-by-step multiplication ensures that every term is correctly accounted for, preserving the balance and integrity of the equation.

Property of Distribution

The Property of Distribution, also known as the Distributive Property, is a key algebraic property that simplifies multiplication over addition and subtraction.
According to the Distributive Property, for any numbers or variables \(a, b,\) and \(c\): \( a(b + c) = ab + ac \) and \( a(b - c) = ab - ac \) This property allows us to break down complex expressions into simpler parts.
In our exercise with \(u(v - 10)\), we distributed \(u\) over both terms inside the parentheses:

• First, \(u\) multiplied by \(v\) gives \(uv\).
• Then, \(u\) multiplied by \(-10\) gives \(-10u\).

This made the expression easier to handle and solve.
Mastering the Distributive Property is essential for simplifying and solving many algebraic expressions and equations.