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Chapter 5: Problem 17

Subtract $$ (2 x+5)-(3 x-9) $$

Short Answer

Expert verified
\(-x + 14\)

Step by step solution

01

Distribute the Negative Sign

To subtract one expression from another, distribute the negative sign across the second expression: \( (2x + 5) - (3x - 9) \) becomes \( 2x + 5 - 3x + 9 \).
02

Combine Like Terms

Now combine the like terms from the expression. The like terms here are \(2x\) and \(-3x\), and the constant terms \(5\) and \(9\). Thus, the expression becomes:\( (2x - 3x) + (5 + 9) = -x + 14 \).

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Key Concepts

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

Subtracting Polynomials
When working with polynomials, subtracting one polynomial from another involves a few straightforward steps. Subtraction is essentially the addition of the opposite. For instance, subtracting \((3x - 9)\) from \((2x + 5)\) means you need to find the difference between these two expressions.
To do this effectively:
  • First, recognize that subtraction can be rephrased as the addition of the additive inverse. This step is crucial because it sets the stage for future operations.
  • In practice, this means you will change each term in the second polynomial to its opposite sign before combining.
  • This ensures all terms are properly prepared for the following steps of simplification.
Subtracting polynomials can seem tricky at first but breaking it into clear, deliberate steps simplifies the process.
Distributive Property
The distributive property is a fundamental aspect of algebra that helps you manage expressions especially when subtracting polynomials. It allows you to "distribute" or spread out any negative signs across an expression. For instance, consider the expression: \((2x + 5) - (3x - 9)\). When you introduce a negative sign in front of a polynomial, it affects every term within parentheses:
  • Convert \((3x - 9)\) into \(-3x + 9\). Here, each term changes its sign.
  • This transformation is critical, as it ensures each term is correctly oriented for addition.
Using the distributive property effectively helps prevent errors when subtracting, especially when larger or more complex polynomials are involved. It shines as a strategy to simplify and prepare expressions for further manipulation.
Combining Like Terms
Combining like terms is a vital skill in algebra that makes expressions more manageable and comprehensible. Like terms are terms whose variables (and their corresponding exponents) are identical. This technique is significant after distributing negative signs across a polynomial.Given the adjusted expression \(2x + 5 - 3x + 9\):
  • Identify like terms such as \(2x\) and \(-3x\), which both contain the variable \(x\).
  • Group and simplify: \((2x - 3x)\) simplifies to \(-x\).
  • Repeat the process with constant terms. The constants \(5\) and \(9\) combine to give \(14\).
The ability to combine like terms transforms complex expressions into simpler ones. This systematic approach ensures clarity and precision, leading to the final simplified result.

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Most popular questions from this chapter

Simplify each expression. Assume that variables represent positive integers. $$ a^{-4 m} \cdot a^{5 m} $$

Subtract using a vertical format. $$ \begin{array}{r} 4 z^{2}-8 z+3 \\ -\left(6 z^{2}+8 z-3\right) \\ \hline \end{array} $$

Add \(\left(4 x^{2}-6 x+1\right)\) and \(\left(3 x^{2}+2 x+1\right)\)

Simplify each expression. Assume that variables represent positive integers. $$ a^{4 m+1} \cdot a^{4} $$

Add or subtract as indicated. $$ \left(x^{2}+2 x y-y^{2}\right)+\left(5 x^{2}-4 x y+20 y^{2}\right) $$
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