Simplify The Expression:${ -\frac{1}{3} X - 12 + \frac{1}{2} X - 6 }$

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities more efficiently. One of the key concepts in simplifying expressions is combining like terms. In this article, we will focus on simplifying the given expression: $-\frac{1}{3} x - 12 + \frac{1}{2} x - 6$. We will break down the steps involved in simplifying the expression and provide a clear explanation of each step.

Understanding Like Terms

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Like terms are terms that have the same variable raised to the same power. In the given expression, we have two terms with the variable x: $-\frac{1}{3} x$ and $\frac{1}{2} x$. These two terms are like terms because they both have the variable x raised to the power of 1.

Combining Like Terms

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To combine like terms, we need to follow the rules of arithmetic operations. When combining like terms, we add or subtract the coefficients of the terms. In this case, we need to add the coefficients of the two like terms: $-\frac{1}{3} x$ and $\frac{1}{2} x$.

Adding Coefficients

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To add the coefficients, we need to find a common denominator. The least common multiple (LCM) of 3 and 2 is 6. We can rewrite the coefficients with a common denominator:

$-\frac{1}{3} x = -\frac{2}{6} x$ $\frac{1}{2} x = \frac{3}{6} x$

Now, we can add the coefficients:

$-\frac{2}{6} x + \frac{3}{6} x = \frac{1}{6} x$

Simplifying the Expression

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Now that we have combined the like terms, we can simplify the expression by adding the constants:

$-\frac{1}{3} x - 12 + \frac{1}{2} x - 6 = \frac{1}{6} x - 18$

Conclusion

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In this article, we simplified the given expression by combining like terms. We followed the rules of arithmetic operations and added the coefficients of the like terms. We also simplified the expression by adding the constants. By following these steps, we can simplify any expression that involves combining like terms.

Example Problems

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Problem 1

Simplify the expression: $2x + 5 - 3x + 2$

Solution

To simplify the expression, we need to combine the like terms. We have two like terms: $2x$ and $-3x$. We can combine these terms by adding their coefficients:

$2x - 3x = -x$

Now, we can simplify the expression by adding the constants:

$2 - 5 = -3$

The simplified expression is: $-x - 3$

Problem 2

Simplify the expression: $4x - 2 + 2x + 1$

Solution

To simplify the expression, we need to combine the like terms. We have two like terms: $4x$ and $2x$. We can combine these terms by adding their coefficients:

$4x + 2x = 6x$

Now, we can simplify the expression by adding the constants:

$-2 + 1 = -1$

The simplified expression is: $6x - 1$

Tips and Tricks

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• When combining like terms, make sure to follow the rules of arithmetic operations.
• Use a common denominator when adding or subtracting fractions.
• Simplify the expression by adding the constants after combining the like terms.

Final Thoughts

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Simplifying expressions is an essential skill in algebra. By combining like terms, we can simplify expressions and make them easier to solve. In this article, we simplified the given expression by combining like terms and provided a clear explanation of each step. We also provided example problems and tips and tricks to help you master the skill of simplifying expressions.

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Frequently Asked Questions

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Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. In the expression $-\frac{1}{3} x - 12 + \frac{1}{2} x - 6$, the terms $-\frac{1}{3} x$ and $\frac{1}{2} x$ are like terms because they both have the variable x raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you need to follow the rules of arithmetic operations. When combining like terms, you add or subtract the coefficients of the terms. In the expression $-\frac{1}{3} x - 12 + \frac{1}{2} x - 6$, you need to add the coefficients of the two like terms: $-\frac{1}{3} x$ and $\frac{1}{2} x$.

Q: What is the common denominator?

A: The common denominator is the least common multiple (LCM) of the denominators of the fractions. In the expression $-\frac{1}{3} x - 12 + \frac{1}{2} x - 6$, the common denominator is 6.

Q: How do I simplify the expression after combining like terms?

A: After combining like terms, you can simplify the expression by adding the constants. In the expression $-\frac{1}{3} x - 12 + \frac{1}{2} x - 6$, you can simplify the expression by adding the constants: $-12$ and $-6$.

Q: Can I simplify an expression with more than two like terms?

A: Yes, you can simplify an expression with more than two like terms. You need to combine the like terms by adding or subtracting their coefficients, and then simplify the expression by adding the constants.

Q: What if I have a fraction with a variable in the denominator?

A: If you have a fraction with a variable in the denominator, you need to find a common denominator to combine the fractions. For example, in the expression $\frac{x}{2} + \frac{3}{4}$, you need to find a common denominator, which is 4.

Q: Can I simplify an expression with negative coefficients?

A: Yes, you can simplify an expression with negative coefficients. You need to follow the rules of arithmetic operations and add or subtract the coefficients of the terms.

Q: What if I have an expression with parentheses?

A: If you have an expression with parentheses, you need to simplify the expression inside the parentheses first, and then combine the like terms.

Example Problems with Answers

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Problem 1

Simplify the expression: $2x + 5 - 3x + 2$

Answer

$-x + 7$

Problem 2

Simplify the expression: $4x - 2 + 2x + 1$

Answer

$6x - 1$

Problem 3

Simplify the expression: $\frac{x}{2} + \frac{3}{4}$

Answer

$\frac{5x + 6}{4}$

Problem 4

Simplify the expression: $-2x + 5 - 3x - 2$

Answer

$-5x + 3$

Tips and Tricks

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• Make sure to follow the rules of arithmetic operations when combining like terms.
• Use a common denominator when adding or subtracting fractions.
• Simplify the expression by adding the constants after combining the like terms.
• If you have a fraction with a variable in the denominator, find a common denominator to combine the fractions.
• If you have an expression with parentheses, simplify the expression inside the parentheses first, and then combine the like terms.

Final Thoughts

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Simplifying expressions is an essential skill in algebra. By combining like terms, you can simplify expressions and make them easier to solve. In this article, we provided a Q&A section to help you understand the concept of combining like terms and simplifying expressions. We also provided example problems with answers to help you practice the skill.