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Chapter 2: Problem 23

Rewrite the expression \(a+2(b-a)-3(c+b)\) without using parentheses. Simplify your answer.

Short Answer

Expert verified
Answer: \(-a - b - 3c\)

Step by step solution

01

Remove parentheses using distributive property

To remove parentheses, we apply the distributive property of multiplication over addition for the expression \(a+2(b-a)-3(c+b)\). This results in \(a + 2b - 2a - 3c - 3b\).
02

Collect like terms

Now let's collect like terms: \begin{align*} &a + 2b - 2a - 3c - 3b \\ =& a - 2a + 2b - 3b - 3c \\ \end{align*}
03

Combine like terms

Combining the like terms, we get: \begin{align*} & (-1)\times a + (-1)\times b - 3c \\ \end{align*} So, the simplified expression after rewriting without parentheses is \(-a - b - 3c\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is all about making mathematical expressions easier to work with by rewriting them in a more straightforward form. To simplify an expression like \(a+2(b-a)-3(c+b)\), involves several steps. First, we need to understand the operations and use algebraic rules to re-organize the expression. These include removing any parentheses, combining like terms, and making calculations clearer.When simplifying, focus on:
  • Removing unnecessary symbols: This means eliminating parentheses through distribution or other algebraic techniques.
  • Rearranging terms: Gather like terms together making the expression more straightforward.
  • Basic arithmetic operations: Perform operations where possible to further reduce complexity.
By following these steps, you can transform complex expressions into a simpler format, which makes solving them easier.
Like Terms
In algebra, like terms refer to terms that contain the same variables raised to the same power. For instance, in the expression \(a + 2b - 2a - 3c - 3b\), like terms are those that involve the same variable. This is crucial because you can only combine like terms through addition or subtraction.Here's a brief breakdown:
  • Identify like terms: Check which terms share the same variable and power. For example, \(a\) and \(-2a\), as well as \(2b\) and \(-3b\).
  • Combine like terms: Add or subtract the coefficients (the numbers before the variables) of these terms. Combining \(a - 2a\) gives \(-a\), and \(2b - 3b\) simplifies to \(-b\).
Understanding like terms allows you to organize and simplify expressions accurately. It helps in solving equations and performing algebraic manipulations more effectively.
Removing Parentheses
Parentheses in algebra often signify operations to be performed first. Removing them simplifies expressions by revealing the true structure. Distributive property is the key rule used here, where multiplication is distributed over addition or subtraction inside the parentheses.In the expression \(a+2(b-a)-3(c+b)\):
  • Apply distributive property: Multiply each term inside the parentheses by the factor directly outside. For \(2(b-a)\), distribute \(2\) to both \(b\) and \(-a\). Similarly, for \(-3(c+b)\), distribute \(-3\).
  • Expand the expression: Once distributed, write the expression without parentheses as \(a + 2b - 2a - 3c - 3b\).
By removing parentheses, expressions like these become simpler, clear, and easier to handle. This skill is vital in algebra to progress from solving basic expressions to tackling more complicated problems.

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