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Chapter 1: Problem 67

Rewrite each expression using the distributive property. Simplify if possible. $$-(v-6)$$

Short Answer

Expert verified
The simplified expression using the distributive property is: \(-(v-6) = -v + 6\).

Step by step solution

01

Identify the distributive property

The distributive property states that for any real numbers a, b, and c, \(a(b + c) = ab + ac\). In our expression, we have a negative sign in front of the parenthesis which is equivalent to a multiplication by -1, so our expression is of the form \(-1(v-6)\).
02

Apply the distributive property

Now we can apply the distributive property to our expression, replacing 'a' with -1, 'b' with 'v', and 'c' with -6: \[-1(v-6) = -1 * v + (-1) * (-6)\]
03

Simplify the expression

We can now simplify the expression by performing the multiplication with -1: \[-1 * v + (-1) * (-6) = -v + 6\] So the simplified expression using the distributive property is: \[-(v-6) = -v + 6\]

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are like sentences in the language of mathematics. They are made up of numbers, variables (which are like placeholders or symbols for numbers), and operations like addition, subtraction, multiplication, and division. In the original exercise, the expression is \( - (v - 6) \), where the variable 'v' is used.
  • Variables represent unknown numbers and allow us to write equations that describe real-world problems.
  • Constants are fixed numbers in an expression, like the number 6 here.
  • Operations describe the action to be performed on the numbers or variables.
Understanding these components is key to manipulating and solving algebraic expressions effectively. When you see an expression with parentheses, such as \( (v-6) \), it means the numbers inside should be handled as a group in calculations, maintaining their relationship to each other.
Simplification
Simplification in algebra involves making an expression easier to work with or understand. One common method is using the distributive property, a powerful algebraic rule. This property lets you multiply a single term by each term within a set of parentheses. In this exercise, we used the distributive property to change \(- (v - 6)\) into \(-v + 6\).
  • Start by identifying what operation is outside the parentheses. Here, it's multiplication by -1.
  • Distribute, or "spread," the multiplication to each term inside the parentheses.
  • Simplify further if possible - here, since each term became simpler, we finalized the expression.
This process of breaking down and simplifying expressions helps in solving equations and understanding relationships between variables and constants easily.
Negative Numbers
Negative numbers represent values less than zero. In algebra, they are handled just like positive numbers but with some key considerations, especially in operations like multiplication and addition. When simplifying an expression like \(- (v - 6)\), the negative sign is multiplied with each term inside the parentheses:
  • Multiplying a negative and a positive results in a negative number - as shown in \(-1 \times v = -v\).
  • Multiplying two negative numbers gives a positive result, seen in \((-1) \times (-6) = +6\).
  • Pay attention to signs; they affect the outcome significantly!
Understanding how negative numbers operate within expressions, and equations are crucial in algebra. This knowledge allows for accurate simplification and problem-solving.

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Most popular questions from this chapter

Write a mathematical expression for each and simplify. 8 more than the quotient of 54 and \(-6\)

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