Simplifying Algebraic Expressions With Mixed Numbers And Fractions

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In the realm of mathematics, simplifying expressions is a fundamental skill that allows us to represent mathematical statements in their most concise and manageable form. This is especially important when dealing with algebraic expressions that involve mixed numbers and fractions. In this comprehensive guide, we will delve into the process of simplifying the expression 1115x−74−3310x+11511 \frac{1}{5} x - \frac{7}{4} - 3 \frac{3}{10} x + 1 \frac{1}{5}, breaking down each step to ensure clarity and understanding. Our focus will be on combining like terms, converting mixed numbers to improper fractions, finding common denominators, and performing the necessary arithmetic operations. Mastering these techniques will not only enhance your ability to simplify expressions but also provide a solid foundation for more advanced mathematical concepts.

At the heart of simplifying expressions lies the concept of combining like terms. Like terms are terms that have the same variable raised to the same power. In our expression, 1115x11 \frac{1}{5} x and −3310x-3 \frac{3}{10} x are like terms because they both contain the variable 'x' raised to the power of 1. Similarly, −74-\frac{7}{4} and 1151 \frac{1}{5} are like terms because they are both constants. By grouping and combining like terms, we can reduce the complexity of the expression. This process makes the expression easier to understand and manipulate, laying the groundwork for further simplification. When simplifying expressions, it is often beneficial to rearrange the terms so that like terms are adjacent to each other. This makes it easier to visually identify and combine them, reducing the chances of error.

Before we can effectively combine like terms, it is essential to convert mixed numbers to improper fractions. A mixed number is a number that consists of a whole number and a proper fraction (where the numerator is less than the denominator). An improper fraction, on the other hand, is a fraction where the numerator is greater than or equal to the denominator. Converting mixed numbers to improper fractions simplifies arithmetic operations such as addition and subtraction. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator of the fraction, add the numerator, and then place the result over the original denominator. For example, to convert 111511 \frac{1}{5} to an improper fraction, we calculate (11×5)+1=56(11 \times 5) + 1 = 56, so the improper fraction is 565\frac{56}{5}. Similarly, 33103 \frac{3}{10} becomes (3×10)+310=3310\frac{(3 \times 10) + 3}{10} = \frac{33}{10}, and 1151 \frac{1}{5} becomes (1×5)+15=65\frac{(1 \times 5) + 1}{5} = \frac{6}{5}. Once all mixed numbers are converted to improper fractions, the expression becomes easier to manipulate and simplify.

Step-by-Step Simplification

To simplify the expression 1115x−74−3310x+11511 \frac{1}{5} x - \frac{7}{4} - 3 \frac{3}{10} x + 1 \frac{1}{5}, we will follow a structured approach, breaking the process down into manageable steps. This methodical approach ensures accuracy and clarity, allowing us to tackle the simplification with confidence.

1. Convert Mixed Numbers to Improper Fractions

The first step in simplifying the expression is to convert all mixed numbers to improper fractions. This will make it easier to perform arithmetic operations later on. We have the mixed numbers 111511 \frac{1}{5}, 33103 \frac{3}{10}, and 1151 \frac{1}{5}. Converting these, we get:

  • 1115=(11×5)+15=56511 \frac{1}{5} = \frac{(11 \times 5) + 1}{5} = \frac{56}{5}
  • 3310=(3×10)+310=33103 \frac{3}{10} = \frac{(3 \times 10) + 3}{10} = \frac{33}{10}
  • 115=(1×5)+15=651 \frac{1}{5} = \frac{(1 \times 5) + 1}{5} = \frac{6}{5}

Substituting these improper fractions into the original expression, we get:

565x−74−3310x+65\frac{56}{5} x - \frac{7}{4} - \frac{33}{10} x + \frac{6}{5}

2. Group Like Terms

Next, we group the like terms together. This involves rearranging the terms so that the terms with 'x' are together and the constant terms are together:

565x−3310x−74+65\frac{56}{5} x - \frac{33}{10} x - \frac{7}{4} + \frac{6}{5}

Grouping like terms makes it visually easier to combine them in the next steps. This organization is a key strategy in simplifying any algebraic expression, ensuring that no terms are missed or incorrectly combined. By focusing on like terms, we can streamline the simplification process and minimize errors.

3. Combine Like Terms with 'x'

Now, we combine the terms with 'x'. To do this, we need to find a common denominator for the fractions 565\frac{56}{5} and 3310\frac{33}{10}. The least common multiple (LCM) of 5 and 10 is 10. So, we convert 565\frac{56}{5} to an equivalent fraction with a denominator of 10:

565=56×25×2=11210\frac{56}{5} = \frac{56 \times 2}{5 \times 2} = \frac{112}{10}

Now we can subtract the fractions:

11210x−3310x=112−3310x=7910x\frac{112}{10} x - \frac{33}{10} x = \frac{112 - 33}{10} x = \frac{79}{10} x

This step demonstrates the importance of finding a common denominator when adding or subtracting fractions. The common denominator allows us to perform the operation on the numerators while keeping the denominator the same. In this case, subtracting the fractions with 'x' results in a single term, 7910x\frac{79}{10} x, which is a significant simplification.

4. Combine Constant Terms

Next, we combine the constant terms, −74-\frac{7}{4} and 65\frac{6}{5}. We need to find a common denominator for these fractions as well. The least common multiple (LCM) of 4 and 5 is 20. So, we convert both fractions to equivalent fractions with a denominator of 20:

  • −74=−7×54×5=−3520-\frac{7}{4} = -\frac{7 \times 5}{4 \times 5} = -\frac{35}{20}
  • 65=6×45×4=2420\frac{6}{5} = \frac{6 \times 4}{5 \times 4} = \frac{24}{20}

Now we can add the fractions:

−3520+2420=−35+2420=−1120-\frac{35}{20} + \frac{24}{20} = \frac{-35 + 24}{20} = -\frac{11}{20}

Combining the constant terms involves the same principle of finding a common denominator. This ensures that the fractions can be accurately added or subtracted. The result of this step is a single constant term, −1120-\frac{11}{20}, which further simplifies the expression.

5. Write the Simplified Expression

Finally, we combine the simplified terms to get the final expression:

7910x−1120\frac{79}{10} x - \frac{11}{20}

This is the simplified form of the original expression. The expression is now in its most concise form, making it easier to work with in further mathematical operations or analyses. The final simplified expression is a combination of a term with the variable 'x' and a constant term, both expressed as fractions.

Final Simplified Expression

The simplified expression is:

7910x−1120\frac{79}{10} x - \frac{11}{20}

This expression represents the original expression in its most simplified form. It is now easier to understand, interpret, and use in further calculations or mathematical manipulations. The process of simplification has reduced the complexity of the expression, making it more accessible and manageable.

Key Concepts and Strategies

Throughout the simplification process, several key concepts and strategies were employed. These are fundamental to simplifying algebraic expressions and are worth highlighting.

Combining Like Terms

Combining like terms is a cornerstone of simplifying algebraic expressions. It involves identifying terms with the same variable raised to the same power and adding or subtracting their coefficients. This reduces the number of terms in the expression and makes it more concise. In our example, we combined the 'x' terms and the constant terms separately, streamlining the expression.

Converting Mixed Numbers to Improper Fractions

Converting mixed numbers to improper fractions is crucial when dealing with arithmetic operations involving mixed numbers. Improper fractions are easier to add, subtract, multiply, and divide. By converting the mixed numbers in our original expression, we set the stage for performing the necessary operations smoothly.

Finding a Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. The common denominator provides a common unit for the fractions, allowing us to perform the operation on the numerators. In our example, we found common denominators for both the 'x' terms and the constant terms, ensuring accurate calculations.

Step-by-Step Approach

Adopting a step-by-step approach is a valuable strategy for simplifying complex expressions. By breaking the process down into manageable steps, we can minimize errors and ensure that each operation is performed correctly. Our step-by-step guide, from converting mixed numbers to writing the final simplified expression, exemplifies this approach.

Common Mistakes to Avoid

Simplifying expressions can be challenging, and it's easy to make mistakes. Being aware of common pitfalls can help you avoid them.

Incorrectly Combining Terms

One common mistake is combining terms that are not like terms. For example, adding an 'x' term to a constant term is incorrect. Always ensure that you are combining terms with the same variable and power.

Errors in Converting Mixed Numbers

Errors in converting mixed numbers to improper fractions can lead to incorrect results. Double-check your calculations to ensure that you have correctly multiplied the whole number by the denominator and added the numerator.

Forgetting the Sign

When combining terms, it's crucial to keep track of the signs (positive or negative) of the terms. A forgotten sign can change the entire result. Pay close attention to the signs and ensure they are correctly carried through the simplification process.

Arithmetic Errors

Basic arithmetic errors, such as incorrect addition or subtraction, can derail the simplification process. Take your time and double-check your calculations to avoid these mistakes.

Not Finding a Common Denominator

Failing to find a common denominator when adding or subtracting fractions is a common error. Remember that fractions must have the same denominator before they can be added or subtracted.

Practice Problems

To solidify your understanding of simplifying expressions, let's work through a few practice problems.

Practice Problem 1

Simplify the expression:

512x+34−214x−1125 \frac{1}{2} x + \frac{3}{4} - 2 \frac{1}{4} x - 1 \frac{1}{2}

Solution:

  1. Convert mixed numbers to improper fractions:

    • 512=1125 \frac{1}{2} = \frac{11}{2}
    • 214=942 \frac{1}{4} = \frac{9}{4}
    • 112=321 \frac{1}{2} = \frac{3}{2}

  2. Substitute the improper fractions into the expression:

    112x+34−94x−32\frac{11}{2} x + \frac{3}{4} - \frac{9}{4} x - \frac{3}{2}

  3. Group like terms:

    112x−94x+34−32\frac{11}{2} x - \frac{9}{4} x + \frac{3}{4} - \frac{3}{2}

  4. Combine the 'x' terms (common denominator is 4):

    • 112x=224x\frac{11}{2} x = \frac{22}{4} x
    • 224x−94x=134x\frac{22}{4} x - \frac{9}{4} x = \frac{13}{4} x

  5. Combine the constant terms (common denominator is 4):

    • 32=64\frac{3}{2} = \frac{6}{4}
    • 34−64=−34\frac{3}{4} - \frac{6}{4} = -\frac{3}{4}

  6. Write the simplified expression:

    134x−34\frac{13}{4} x - \frac{3}{4}

Practice Problem 2

Simplify the expression:

823−416x+213x−568 \frac{2}{3} - 4 \frac{1}{6} x + 2 \frac{1}{3} x - \frac{5}{6}

Solution:

  1. Convert mixed numbers to improper fractions:

    • 823=2638 \frac{2}{3} = \frac{26}{3}
    • 416=2564 \frac{1}{6} = \frac{25}{6}
    • 213=732 \frac{1}{3} = \frac{7}{3}

  2. Substitute the improper fractions into the expression:

    263−256x+73x−56\frac{26}{3} - \frac{25}{6} x + \frac{7}{3} x - \frac{5}{6}

  3. Group like terms:

    −256x+73x+263−56-\frac{25}{6} x + \frac{7}{3} x + \frac{26}{3} - \frac{5}{6}

  4. Combine the 'x' terms (common denominator is 6):

    • 73x=146x\frac{7}{3} x = \frac{14}{6} x
    • −256x+146x=−116x-\frac{25}{6} x + \frac{14}{6} x = -\frac{11}{6} x

  5. Combine the constant terms (common denominator is 6):

    • 263=526\frac{26}{3} = \frac{52}{6}
    • 526−56=476\frac{52}{6} - \frac{5}{6} = \frac{47}{6}

  6. Write the simplified expression:

    −116x+476-\frac{11}{6} x + \frac{47}{6}

Conclusion

Simplifying expressions involving mixed numbers and fractions is a fundamental skill in mathematics. By mastering the techniques of converting mixed numbers to improper fractions, finding common denominators, combining like terms, and following a structured approach, you can confidently tackle complex expressions. Remember to avoid common mistakes and practice regularly to solidify your understanding. With these skills in hand, you'll be well-equipped to tackle more advanced mathematical concepts and problems. The ability to simplify expressions is not just about finding the right answer; it's about developing a deep understanding of mathematical principles and building a strong foundation for future learning.