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Chapter 2: Problem 6

Simplify each expression as completely as possible. $$-4(2 r-3 s)+6(s-r)$$

Short Answer

Expert verified
Simplified expression is -14r + 18s

Step by step solution

01

Distribute the constants

Apply the distributive property to both terms of the expression. For -4(2r-3s)+6(s-r): Distribute -4 across the terms ( 2r - 3s), then distribute 6 across (s-r).
02

Conduct the distribution

First: -4(2r) = -8r -4(-3s)= +12s =6(s-r).
03

Expand second term

Expand the second term in the equation 6(s-r): 6(s) ) =6s - -6(r)-6r.
04

Combine like terms

Combine like terms: -8r -6r = -14r and 12s + 6s= +18s.

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Key Concepts

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

distributive property
The distributive property is very useful when you're working to simplify algebraic expressions. This property states that you can distribute a multiplication over an addition or subtraction within parentheses.

For example, in the given exercise -4(2r - 3s) + 6(s - r), you need to distribute -4 to both 2r and -3s. This process is similar to multiplying -4 by each term inside the parentheses:
  • -4(2r) = -8r
  • -4(-3s) = +12s


Next, do the same for 6(s - r):
  • 6(s) = 6s
  • 6(-r) = -6r


Distributing makes it easier to combine like terms, which is the next concept we'll explore.
combining like terms
Combining like terms is another essential step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power.

In the expression from the exercise, after applying the distributive property, we have:
  • -8r + 12s + 6s - 6r


To combine like terms, group them together:
  • Combine the 'r' terms: -8r - 6r = -14r
  • Combine the 's' terms: 12s + 6s = 18s


After combining, the expression becomes -14r + 18s. This step simplifies the expression by reducing the number of terms.
algebraic simplification
Algebraic simplification involves a set of processes aimed at making an expression easier to understand or work with. The processes we've just discussed, distributing and combining like terms, are crucial steps.

Generally, simplification makes it easier to see the relationships between terms and simplifies solving equations or inequalities.

Here's a summary of the simplification process seen in this exercise:
  • Distribute constants: -4(2r - 3s) + 6(s - r) becomes -8r + 12s + 6s - 6r
  • Combine like terms: -8r - 6r = -14r and 12s + 6s = 18s


The final expression is simplified to -14r + 18s. By following these steps, you've taken a more complicated expression and made it more manageable!

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Most popular questions from this chapter

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