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Chapter 3: Problem 39

Simplify the expression. $$-(-5 x+1+9 y)$$

Short Answer

Expert verified
The simplified expression is \(5x - 1 - 9y\).

Step by step solution

01

Understand the Problem

We need to simplify the given expression: \(-(-5x + 1 + 9y)\). This means removing the negative sign from outside the parentheses by multiplying it with each term inside the parentheses.
02

Apply the Distributive Property

To remove the negative sign in front of the parentheses, apply the distributive property. This involves multiplying \(-1\) with each term inside the parentheses: \(-1\cdot(-5x)\), \(-1\cdot 1\), and \(-1\cdot 9y\).
03

Simplify Each Term

Calculate each term from Step 2:- \(-1\cdot(-5x) = 5x\)- \(-1\cdot 1 = -1\)- \(-1\cdot 9y = -9y\).
04

Combine the Simplified Terms

Write the simplified terms together as a single expression: \(5x - 1 - 9y\). This is the simplified form of the expression.

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a core concept in algebra that deals with distributing a factor over terms inside parentheses. This property allows you to multiply each term independently with the factor outside the parentheses. In our exercise, the factor is \(-1\). Here's how it works:
  • Take the factor outside the parenthesis, in this case, \(-1\).
  • Multiply it by each term inside the parentheses.
  • Remember that multiplication can affect the sign as well.
Applying the distributive property is essential when simplifying algebraic expressions, especially when dealing with negative numbers. It helps to break down the expression into simpler terms.
The Role of a Negative Sign
Handling negative signs can be tricky, but it's crucial to understand their impact when simplifying expressions. When you see a negative sign, like in our problem \(-(-5x + 1 + 9y)\), the sign indicates that you need to change each term's sign inside the parentheses.
  • A negative multiplied by a negative becomes a positive (e.g., \(-1 \, \cdot \, -5x = 5x\)).
  • A negative multiplied by a positive remains a negative (e.g., \(-1 \, \cdot \, 9y = -9y\)).
  • A negative multiplied by zero stays zero.
Managing negative signs correctly ensures accuracy in your calculations and leads to the correct simplified form of the expression.
Combining Like Terms
Once you have simplified each term using the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable parts raised to the same power. For example, terms with \(x\) can be combined.In our simplified expression \(5x - 1 - 9y\):
  • The term \(5x\) stands alone because there are no other \(x\) terms to combine it with.
  • The constant \(-1\) is also unique, as there are no other numerical constants.
  • The term \(-9y\) is the only term with \(y\), so it stays as it is.
Combining like terms streamlines expressions, making them easier to evaluate and understand. It鈥檚 the last step to tidying up an expression before reaching its simplest form.

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