Simplify: $\frac{2}{3} A + 6 - \frac{3}{5} A - 1$A) $\frac{1}{15} A + 5$ B) $-\frac{1}{15} A + 5$ C) $-\frac{2}{5} A + 7$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying a specific algebraic expression, 23a+6−35a−1\frac{2}{3} a + 6 - \frac{3}{5} a - 1, and explore the different methods and techniques used to achieve this goal.

Understanding the Expression

The given expression is 23a+6−35a−1\frac{2}{3} a + 6 - \frac{3}{5} a - 1. To simplify this expression, we need to combine like terms and eliminate any unnecessary components. The expression consists of two fractions, 23a\frac{2}{3} a and −35a-\frac{3}{5} a, and two constants, 66 and −1-1.

Step 1: Combine Like Terms

The first step in simplifying the expression is to combine like terms. In this case, we have two fractions with different denominators, 23a\frac{2}{3} a and −35a-\frac{3}{5} a. To combine these fractions, we need to find a common denominator, which is the least common multiple (LCM) of the two denominators.

Finding the Least Common Multiple (LCM)

The LCM of 33 and 55 is 1515. Therefore, we can rewrite the fractions with the common denominator of 1515.

23a=2×53×5a=1015a\frac{2}{3} a = \frac{2 \times 5}{3 \times 5} a = \frac{10}{15} a

−35a=−3×35×3a=−915a-\frac{3}{5} a = -\frac{3 \times 3}{5 \times 3} a = -\frac{9}{15} a

Combining the Fractions

Now that we have the fractions with the common denominator, we can combine them by adding or subtracting the numerators.

1015a−915a=10−915a=115a\frac{10}{15} a - \frac{9}{15} a = \frac{10 - 9}{15} a = \frac{1}{15} a

Simplifying the Constants

The next step is to simplify the constants, 66 and −1-1. We can combine these constants by adding or subtracting them.

6−1=56 - 1 = 5

Final Simplified Expression

Now that we have combined the like terms and simplified the constants, we can write the final simplified expression.

115a+5\frac{1}{15} a + 5

Conclusion

In this article, we have simplified the algebraic expression 23a+6−35a−1\frac{2}{3} a + 6 - \frac{3}{5} a - 1 by combining like terms and eliminating unnecessary components. We have used the concept of the least common multiple (LCM) to find a common denominator for the fractions and then combined the fractions by adding or subtracting the numerators. Finally, we have simplified the constants by adding or subtracting them. The final simplified expression is 115a+5\frac{1}{15} a + 5.

Answer Options

Based on the simplified expression, we can conclude that the correct answer is:

A) 115a+5\frac{1}{15} a + 5

Discussion

This problem requires students to apply their knowledge of algebraic expressions and simplification techniques. The problem is designed to test the student's ability to combine like terms, find a common denominator, and simplify constants. The solution requires a step-by-step approach, and the student must carefully follow the instructions to arrive at the correct answer.

Tips and Tricks

  • When simplifying algebraic expressions, it is essential to combine like terms and eliminate unnecessary components.
  • To find a common denominator, use the concept of the least common multiple (LCM).
  • When combining fractions, add or subtract the numerators.
  • When simplifying constants, add or subtract them.

Practice Problems

  1. Simplify the expression 45x+2−35x−1\frac{4}{5} x + 2 - \frac{3}{5} x - 1.
  2. Simplify the expression 23y−4+13y+2\frac{2}{3} y - 4 + \frac{1}{3} y + 2.
  3. Simplify the expression 34z+1−14z−3\frac{3}{4} z + 1 - \frac{1}{4} z - 3.

Answer Key

  1. 15x+1\frac{1}{5} x + 1
  2. 33y−2\frac{3}{3} y - 2
  3. 24z−2\frac{2}{4} z - 2

Conclusion

Introduction

In our previous article, we explored the concept of simplifying algebraic expressions and provided a step-by-step guide on how to simplify the expression 23a+6−35a−1\frac{2}{3} a + 6 - \frac{3}{5} a - 1. In this article, we will provide a Q&A guide to help students and professionals alike understand the concept of simplifying algebraic expressions.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: Why is it important to simplify algebraic expressions?

A: Simplifying algebraic expressions is essential because it helps to:

  • Reduce the complexity of the expression
  • Make it easier to solve and manipulate
  • Identify patterns and relationships between variables
  • Improve understanding and interpretation of the expression

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, 2x2x and 3x3x are like terms because they both have the variable xx raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, 2x+3x=(2+3)x=5x2x + 3x = (2 + 3)x = 5x.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 3 and 5 is 15.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that they have in common. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. For example, the GCD of 12 and 15 is 3.

Q: How do I simplify constants?

A: To simplify constants, you can add or subtract them. For example, 2+3=52 + 3 = 5.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not combining like terms
  • Not finding the LCM of fractions
  • Not simplifying constants
  • Not checking for errors in the expression

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

  • Working through examples and exercises
  • Using online resources and tools
  • Asking a teacher or tutor for help
  • Joining a study group or online community

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for students and professionals alike. By understanding the concept of like terms, LCM, and GCD, and by practicing simplifying algebraic expressions, you can improve your understanding and interpretation of mathematical expressions. Remember to avoid common mistakes and to seek help when needed.

Practice Problems

  1. Simplify the expression 45x+2−35x−1\frac{4}{5} x + 2 - \frac{3}{5} x - 1.
  2. Simplify the expression 23y−4+13y+2\frac{2}{3} y - 4 + \frac{1}{3} y + 2.
  3. Simplify the expression 34z+1−14z−3\frac{3}{4} z + 1 - \frac{1}{4} z - 3.

Answer Key

  1. 15x+1\frac{1}{5} x + 1
  2. 33y−2\frac{3}{3} y - 2
  3. 24z−2\frac{2}{4} z - 2

Additional Resources

  • Khan Academy: Algebraic Expressions
  • Mathway: Algebraic Expressions
  • Wolfram Alpha: Algebraic Expressions