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Chapter 12: Problem 2

Add \((12 x+6 y)+(2 x-7 y)\) A. \(10 x+10 y\) B. \(14 x-y\) C. \(10 x-6 y\) D. \(14 x+y\)

Short Answer

Expert verified
The correct answer is B: \(14x-y\).

Step by step solution

01

Identify Like Terms

To combine two algebraic expressions, we first identify like terms. The like terms in the expressions \((12x + 6y)\) and \((2x - 7y)\) are: - Terms with \(x\): \(12x\) and \(2x\)- Terms with \(y\): \(6y\) and \(-7y\).
02

Add the Like Terms

Now that we have identified the like terms, we can add them:- For the \(x\) terms: \(12x + 2x = 14x\)- For the \(y\) terms: \(6y - 7y = -1y\) or \(-y\).
03

Write the Simplified Expression

Combine the results from Step 2 to form the simplified expression:\[14x - y\].
04

Find the Matching Answer Choice

The simplified expression \(14x - y\) is option B. Therefore, \( (12x + 6y) + (2x - 7y) = 14x - y \).

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Key Concepts

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

Combining Like Terms
When working with algebraic expressions, one of the essential skills is combining like terms. Like terms have the same variable raised to the same power. For example, in our exercise, you can see terms like \(12x\) and \(2x\) or \(6y\) and \(-7y\). These are considered like terms because:
  • They both contain the variable \(x\) or \(y\)
  • They are each raised to the first power
To combine like terms, simply add or subtract their coefficients (the numerical part of the term in front of the variable). In this exercise:
  • Combine \(12x\) and \(2x\) to get \(14x\)
  • Combine \(6y\) and \(-7y\) to get \(-y\)
By following this approach, we can simplify expressions and make them easier to work with.
Simplifying Expressions
Simplifying expressions is all about making your algebraic expressions as concise as possible. The goal is to reduce the expression to its simplest form without changing its value. Let's look at the simplified expression we achieved in the exercise.

The original problem was \((12x + 6y) + (2x - 7y)\). After combining the like terms, we obtained \(14x - y\). Why is this considered simpler?

  • Fewer terms make it easier to understand and work with
  • It makes it clearer how different variables interact within the expression
  • Less clutter means fewer opportunities for error in calculations
By practicing simplification, you enhance your ability to tackle more complex algebraic problems later on.
Algebraic Expressions
Algebraic expressions are combinations of letters and numbers combined by operations such as addition, subtraction, multiplication, and division. In our exercise, the parts \(12x + 6y\) and \(2x - 7y\) are examples of these expressions. Here's how they work:
  • The letters (like \(x\) and \(y\)) are variables. They represent numbers that can change or vary.
  • The numbers in front of the variables (like \(12\), \(6\), \(2\), and \(-7\)) are called coefficients. They tell us how many of each variable we have.
  • Operations such as addition and subtraction connect these parts into a single expression.
Being comfortable working with algebraic expressions forms the foundation of all algebraic problem-solving, enabling everything from simple calculations to high-level equations.

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Most popular questions from this chapter

Add \(\left(4 x^{2}+2 x-1\right)+\left(9 x^{2}-8 x-3\right)\) A. \(13 x^{2}-4 x+2\) B. \(6 x^{2}-4 x+2\) c. \(5 x^{2}-5 x+2\) D. \(13 x^{2}-6 x-4\)

Divide \(-42 a^{5} b^{3} c^{4} \div 2 a^{3} b^{2} c\) A. \(-21 a^{2} b c^{3}\) B. \(21 a^{2} b^{2} c\) C. \(-21\) abc \(^{2}\) D. \(21 a^{2} b^{2} c^{3}\)

Find \(\left(4 x^{2}\right)^{4}\) A. \(256 x^{8}\) B. \(64 x^{6}\) C. \(12 x^{6}\) D. \(16 x^{8}\)

Multiply \(y\left(3 y^{3}-y+8\right)\) A. \(3 y^{4}-y^{2}+8 y\) B. \(3 y^{3}+y^{2}+8 y\) C. \(y^{4}+y^{2}+8\) D. \(3 y^{3}-y+8 y\)

Find \(\left(-3 x^{3} y^{2} x\right)^{4}\) A. \(-81 x^{12} y^{8}\) B. \(12 x^{7} y^{6}\) C. \(-12 x^{7} y^{6}\) D. \(81 x^{12} y^{8}\)
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