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Chapter 1: Problem 88

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

Short Answer

Expert verified
10u - 6v + 14w

Step by step solution

01

- 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.
02

- 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).
03

- Multiply the terms

Carry out the multiplication for each term: - 2 * 5u = 10u - 2 * -3v = -6v - 2 * 7w = 14w.
04

- Combine the terms

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve each problem. The surface, or rim, of a canyon is at altitude \(0 .\) On a hike down into the canyon, a party of hikers stops for a rest at \(130 \mathrm{~m}\) below the surface. The hikers then descend another \(54 \mathrm{~m}\). Write the new altitude as a signed number.

Solve each problem. In 2005, Americans saved \(-0.5 \%\) of their after-tax incomes. In September 2017 , they saved \(3.1 \%\). (Data from U.S. Bureau of Economic Analysis.) (a) Express the difference between these amounts as a positive number. (b) How could Americans have a negative personal savings rate in \(2005 ?\)

Write each sentence as an equation, using \(x\) as the variable. Then find the solution of the equation from the set of integers between -12 and \(12,\) inclusive. An integer is divisible by 8 if its last three digits form a number divisible by \(8,\) and not otherwise. (a) Is 2,923,296 divisible by \(8 ?\) (b) Is 7,291,623 divisible by \(8 ?\)

Use the distributive property to rewrite each expression. $$ 8(3 r+4 s-5 y) $$

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \((5 x-2 y)(-2 a)\)
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