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Chapter 1: Problem 37

$$\text { Simplify the given algebraic expressions.}$$ $$5 p-(q-2 p)-[3 q-(p-q)]$$

Short Answer

Expert verified
The simplified expression is \(8p - 5q\).

Step by step solution

01

Distribute the Negative Sign in the Parentheses

Start by distributing the negative signs in the expressions \( (q-2p) \) and \( (p-q) \). This will give us: \[ 5p - q + 2p - [3q - p + q] \]
02

Simplify Expressions Inside the Brackets

Next, simplify the expression inside the brackets by combining like terms. Replace \( 3q - p + q \) with \( 4q - p \): \[ 5p - q + 2p - [4q - p] \]
03

Distribute the Negative Sign in the Brackets

Distribute the negative sign in front of the bracket \( [4q - p] \): \[ 5p - q + 2p - 4q + p \]
04

Combine Like Terms

Combine the like terms to simplify the expression: Combine \( 5p, 2p, \text{ and } p \) to get \( 8p \).Combine \( -q \) and \( -4q \) to get \( -5q \).Thus, the simplified expression is: \[ 8p - 5q \]

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Key Concepts

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

Simplifying Expressions
Simplifying algebraic expressions is a way to make complex expressions easier to work with. This is achieved by performing operations and reducing expressions to their simplest form. Simplification is essential in solving algebra problems, as it helps present the solution more clearly and efficiently.
To simplify expressions, start by identifying all operations that can be performed. This includes distributing signs, combining like terms, and eliminating any unnecessary terms or factors.
  • Look for terms that can be combined or transformed to reduce the overall number of terms.
  • Consider order of operations鈥攈andle anything inside parentheses or brackets first.
  • Consistently simplify step by step, ensuring accuracy at each stage.
Following these steps ensures that any algebraic expression can be distilled down to its simplest and most manageable form.
Distributing Negative Signs
Distributing negative signs correctly is a crucial step in simplifying algebraic expressions. When you encounter a negative sign in front of a set of parentheses, it affects all the terms inside. Let鈥檚 go through this essential process:
Suppose you have the expression: a - (b - c)
To distribute the negative sign:
  • Change the signs of all terms inside the parentheses.
  • This results in: a - b + c.
Sometimes negative signs are tricky, but remember they flip the sign of every term within the parentheses. In our original problem:
  • When distributing the negative through (q-2p), the expression becomes -q + 2p.
  • For the brackets [3q-(p-q)], you'll distribute to get -4q + p afterwards.
This process simplifies handling expressions significantly by eliminating the nested expressions.
Combining Like Terms
Combining like terms is the final touch in the simplification process. Like terms are those that have identical variable parts.
For example, in the expression 5p + 2p - q + 3q:
  • 5p and 2p are like terms.
  • -q and 3q are like terms.
Combine them by adding or subtracting their coefficients:
  • 5p + 2p gives you 7p.
  • -q + 3q gives you 2q.
For the given exercise, combining like terms in 5p + 2p - q - 4q + p results in:
  • Adding up 5p, 2p, and p gives you 8p.
  • Combining -q and -4q gives you -5q.
Thus, the simplified version of the expression is 8p - 5q. This step helps in reducing the expression further, making it less cumbersome and easier to work with.

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