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Chapter 7: Problem 131

In the following exercises, simplify using the distributive property. $$-4-(c-10)$$

Short Answer

Expert verified
- c + 6

Step by step solution

01

Identify the Expression to Distribute

The given expression is equation . We need to distribute the term inside the parentheses.
02

Apply the Distributive Property

Use the distributive property which states that a(b+c) = ab + ac . Distribute the negative sign inside the parentheses: formula .
03

Simplify the Expression

Combine like terms if necessary. In this case, we first distribute the negative sign: eqn(-4 - c + 10) . Next, we combine the constant terms: eqn(-4 + 10 - c) which simplifies to .

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Key Concepts

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

Simplify Expressions
Simplifying expressions is an essential part of algebra. It involves reducing an expression to its most basic form without changing its value. This can make it easier to work with and understand.
Here are some general steps to follow:
  • Remove any parentheses by distributing or combining terms inside the parentheses.
  • Combine like terms, which are terms with the same variables raised to the same power.
  • Arrange the terms in a standard order, usually with terms involving variables first and constants last.
For example, in the expression \(-4 - (c - 10)\), we first need to distribute the negative sign. This small step can greatly clarify the overall structure of the problem, making further actions simpler.
Combine Like Terms
Combining like terms is another important concept in algebra. Like terms are terms that have the same variables and the same powers. For instance, \(5x\) and \(-3x\) are like terms because they both include the variable \(x\) raised to the same power of 1.
When combining like terms, you simply add or subtract the coefficients (the numbers in front of the variables) and keep the variable part unchanged. Here鈥檚 a step-by-step example using our exercise:
  • We have the expression \( -4 - c + 10 \), which includes constant terms \(-4\) and \(10\).
  • Combine \( -4 + 10 \): It simplifies to \( 6 \).
  • The result of simplifying and combining like terms gives \( 6 - c \).
Now, the expression is easier to work with. Always remember to look for like terms if you want to simplify expressions correctly.
Negative Numbers
Working with negative numbers can be tricky, but it鈥檚 crucial to understand how to handle them properly in algebra.
Negative numbers have some special rules:
  • When you multiply or divide two negative numbers, the result is positive. For example, \(-2 \times -3 = 6\).
  • When you multiply or divide a positive number by a negative number, the result is negative. For example, \(-2 \times 3 = -6\).
  • Adding a negative number is the same as subtracting that number: \(-4 + (-6) = -10\).
In the context of our exercise, distributing the negative sign involves using these rules:
  • If you start with \( -4 - (c - 10) \), you distribute the negative sign to both \( c \) and \( -10 \).
  • This converts the expression to \( -4 - c + 10 \).
Here, we use the rule that subtracting a negative turns it into a positive. Understanding these rules helps you avoid common mistakes and simplifies working with negative numbers.

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