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Chapter 5: Problem 65

Simplify. $$ (2 x-7)-(x 2+3 x-7)+(6 x-1) $$

Short Answer

Expert verified
The simplified expression is \(-x^2 + 5x - 1\).

Step by step solution

01

Distribute Minus Sign

Begin by distributing the minus sign in front of the middle expression \((x^2 + 3x - 7)\). The expression becomes: \((2x - 7) - x^2 - 3x + 7 + (6x - 1)\).
02

Rearrange the Expression

Rearrange the terms to group similar types together. They can be rearranged as \(-x^2 + 2x - 3x + 6x - 7 + 7 - 1\).
03

Combine Like Terms

Start by combining the \(x\) terms: \(2x - 3x + 6x = 5x\). Then, combine the constant terms: \(-7 + 7 - 1 = -1\). The expression simplifies to \(-x^2 + 5x - 1\).
04

Final Simplified Expression

The fully simplified version of the expression is \(-x^2 + 5x - 1\), as there are no more like terms to combine.

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Key Concepts

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

Distributing Operations in Algebra
The first step in simplifying algebraic expressions often involves distributing operations, like negative signs, correctly across the terms. In our expression
  • oindent oindent (2x - 7) - (x^2 + 3x - 7) + (6x - 1) we need to distribute the minus sign in front of the middle expression, \(x^2 + 3x - 7\).
  • To do this, you apply the minus sign to each term inside the parentheses, effectively changing their signs. This transforms:
    • \(x^2\) into \(-x^2\),
    • \(+3x\) into \(-3x\), and
    • \(+7\) into \(-7\).
  • The expression now looks like \(2x - 7 - x^2 - 3x + 7 + 6x - 1\).Distributing might seem simple, but it's crucial for ensuring all terms are correctly included in further steps of simplification.
Combining Like Terms
Once distribution is completed, it's time to tidy up the expression by combining like terms.
  • 'Like terms' include those with the same variable raised to the same power. If we look at the expression after distributing, we see:
    • Terms with \(x^2\): \(-x^2\)
    • Terms with \(x\): \(2x - 3x + 6x\)
    • Constant terms: \(-7 + 7 - 1\).
  • When you combine an expression's like terms, you're grouping them based on their variables and constants:
    • For \(x\) terms, add coefficients: \(2 - 3 + 6 = 5\) so it becomes \(5x\).
    • For constants: \(-7 + 7 - 1 = -1\).
    The expression simplifies further to \(-x^2 + 5x - 1\). Combining like terms helps consolidate the expression, making it easier to interpret and use in equations.
    • Algebraic Expression Simplification
    The final step in handling algebraic expressions is simplification, which wraps up the process by ensuring the expression is as straightforward as possible.
  • After distribution and combining like terms, our expression is \(-x^2 + 5x - 1\).
  • At this stage, your goal is to ensure:
    • There are no further terms that can be simplified.
    • There are no parentheses or unnecessary terms in the expression.
  • Check each part of the expression to confirm that it cannot be reduced further. Here, there's only one term each for \(-x^2\), \(5x\), and the constant \(-1\).
  • By finalizing the expression like this, you ensure that the expression is in its most efficient form, ready for use in equations or further mathematical processes.

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