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Chapter 10: Problem 72

Perform the indicated operations. $$ (12+7 y)-(-3+2 y) $$

Short Answer

Expert verified
15 + 5y

Step by step solution

01

Distribute the Negative Sign

First, distribute the negative sign across the second group of terms: (12 + 7y) - (-3 + 2y) becomes (12 + 7y) - 1*(-3) - 1*(2y).
02

Simplify the Expression

Now, simplify the expression by distributing the negative sign: (12 + 7y) - (-3) - (2y) This simplifies to: (12 + 7y + 3 - 2y).
03

Combine Like Terms

Lastly, combine like terms: 12 + 3 is 15 and 7y - 2y is 5y. So, the final expression is 15 + 5y.

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Key Concepts

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

Distributing Negative Signs
When you see a negative sign in front of parentheses, it means you need to distribute it across all the terms inside the parentheses. This is like multiplying each term inside the parentheses by -1. For example, if you have (12 + 7y) - (-3 + 2y), you distribute the negative sign like this:
  • (12 + 7y) - 1*(-3) - 1*(2y)
What does this do?
  • -(-3) becomes +3 because -1 times -3 is 3.
  • -(2y) becomes -2y because -1 times 2y is -2y.
So, after distributing the negative sign, the expression looks like this:
  • (12 + 7y) + 3 - 2y
Combining Like Terms
Combining like terms means to group and combine terms that have the same variables to simplify your expression. In our case, consider these terms separately:
  • Constants: 12 and 3
  • Variable terms: 7y and -2y
You can only combine terms that have the same variable. Here, combine the constants first:
  • 12 + 3 = 15
Then, combine the variable terms:
  • 7y - 2y = 5y
Finally, the expression now looks like this:
  • 15 + 5y
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. It makes the expression easier to work with. Here, we simplified (12 + 7y) - (-3 + 2y) through two main steps:
  • Distributing the negative sign: This means converting -(-3) to +3 and -(2y) to -2y.
  • Combining like terms: Group constants together and variable terms together. So, 12 + 3 becomes 15 and 7y - 2y becomes 5y.
After performing these steps, the simplified expression is:
  • 15 + 5y
Remember, simplifying makes it easier to substitute values or solve the equation later. Always follow these steps carefully to avoid errors.

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Most popular questions from this chapter

Find each quotient. $$ \frac{3+i}{i} $$

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Concept Check Which sum could be simplified without first simplifying the individual radical expressions? A. \(\sqrt{50}+\sqrt{32}\) B. \(3 \sqrt{6}+9 \sqrt{6} \quad\) C. \(\sqrt[3]{32}+\sqrt[3]{108}\) D. \(\sqrt[5]{6}+\sqrt[5]{192}\)

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