Problem 88 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ 2(5 u-3 v+7 w) $$

Step-by-Step Solution

Answer

10u - 6v + 14w

Step 1 - Identify the distributive property

The distributive property states that a(b + c + d) = ab + ac + ad. This means you distribute the outside term to each term inside the parentheses.

Step 2 - Apply the distributive property

Here, we have 2(5u - 3v + 7w). Distribute 2 to each term inside the parentheses: (2 * 5u), (2 * -3v), and (2 * 7w).

Step 3 - Multiply the terms

Carry out the multiplication for each term: - 2 * 5u = 10u - 2 * -3v = -6v - 2 * 7w = 14w.

Step 4 - Combine the terms

Combine the results of your multiplications to get the final expression: 10u - 6v + 14w.

Key Concepts

algebraic expressions

Understanding algebraic expressions is key in algebra. An algebraic expression consists of numbers, variables, and mathematical operations. For example, in the expression \(5u - 3v + 7w\), the numbers 5, 3, and 7 are coefficients, and \(u, v,\) and \(w\) are variables.
These expressions do not contain an equals sign; thus, they do not represent complete equations.

• Terms: Each part of the expression that is added or subtracted, like \(5u\) or \(-3v\).
• Coefficients: The numbers multiplying the variables. In \(5u\), 5 is the coefficient.
• Constants: Numbers without variables. In our example, there are no constants.

Breaking down expressions makes them easier to manipulate, such as when using the distributive property.

multiplication

Multiplication in algebra involves variables and numbers. When applying the distributive property, you multiply the term outside the parenthesis by each term inside. Let's look at the exercise: \(2(5u - 3v + 7w)\).
Here's how it works step-by-step:

• First, identify the term outside the parenthesis: 2.
• Next, multiply 2 by each term inside: \(5u, -3v,\) and \(7w\).
• This results in: \(2 * 5u = 10u\), \(2 * -3v = -6v\), and \(2 * 7w = 14w\).

The multiplication step is crucial for simplifying expressions and solving equations efficiently. Always distribute the multiplying term carefully to avoid errors.

combining like terms

Combining like terms means simplifying expressions by merging terms with the same variables. In our example, \(10u - 6v + 14w\), each term is already simplified. But understanding how to combine like terms is essential.
Here's what you need to know:

• Identify like terms: Terms with the same variable, like \(10u\) and \(15u\).
• Add or subtract the coefficients of these like terms. For example, \(10u + 15u = 25u\).
• Ensure the variables and their exponents are the same to combine the terms correctly.

In more complex expressions, this step simplifies the problem, making it easier to solve. Remember, practice makes perfect, so keep working on recognizing and combining like terms.