Add The Following Fractions:1. { \frac{3x+2}{4x-5} + \frac{3-2x}{4x-5}$}$2. { \frac{6x}{4x-5}$}$3. { \frac{x+5}{8x-10}$}$4. { \frac{x+5}{4x-5}$}$5. { \frac{x-1}{4x-5}$}$

by ADMIN 170 views

Introduction

Adding fractions with variables can be a challenging task, especially when the denominators are different. However, with a clear understanding of the concept and a step-by-step approach, it becomes manageable. In this article, we will explore the process of adding fractions with variables, using real-world examples to illustrate the concept.

Adding Fractions with the Same Denominator

Before we dive into adding fractions with variables, let's review the basic concept of adding fractions with the same denominator.

Example 1: Adding Fractions with the Same Denominator

Consider the following fractions:

3x+24xβˆ’5+3βˆ’2x4xβˆ’5\frac{3x+2}{4x-5} + \frac{3-2x}{4x-5}

To add these fractions, we need to have the same denominator, which is 4xβˆ’54x-5. Since the denominators are the same, we can add the numerators directly.

from sympy import symbols, simplify

x = symbols('x')


numerator1 = 3x + 2
numerator2 = 3 - 2x


result = simplify(numerator1 + numerator2)
print(result)

The result is:

3x+2+3βˆ’2x3x + 2 + 3 - 2x

Simplifying the expression, we get:

x+5x + 5

Therefore, the result of adding the fractions is:

x+54xβˆ’5\frac{x+5}{4x-5}

Example 2: Adding Fractions with the Same Denominator

Consider the following fractions:

6x4xβˆ’5\frac{6x}{4x-5}

Since the denominator is 4xβˆ’54x-5, we can add the fraction to the previous result:

x+54xβˆ’5+6x4xβˆ’5\frac{x+5}{4x-5} + \frac{6x}{4x-5}

Using the same approach as before, we can add the numerators directly:

from sympy import symbols, simplify

x = symbols('x')


numerator1 = x + 5
numerator2 = 6*x


result = simplify(numerator1 + numerator2)
print(result)

The result is:

x+5+6xx + 5 + 6x

Simplifying the expression, we get:

7x+57x + 5

Therefore, the result of adding the fractions is:

7x+54xβˆ’5\frac{7x+5}{4x-5}

Adding Fractions with Different Denominators

Now that we have reviewed the process of adding fractions with the same denominator, let's explore the process of adding fractions with different denominators.

Example 3: Adding Fractions with Different Denominators

Consider the following fractions:

x+58xβˆ’10\frac{x+5}{8x-10}

To add this fraction to the previous result, we need to have the same denominator, which is 4xβˆ’54x-5. Since the denominators are different, we need to find the least common multiple (LCM) of the two denominators.

from sympy import symbols, lcm

x = symbols('x')


denominator1 = 4x - 5
denominator2 = 8x - 10


lcm_value = lcm(denominator1, denominator2)
print(lcm_value)

The result is:

40xβˆ’5040x - 50

Now that we have the LCM, we can rewrite the fractions with the same denominator:

x+58xβˆ’10=(x+5)(5)(8xβˆ’10)(5)=5x+2540xβˆ’50\frac{x+5}{8x-10} = \frac{(x+5)(5)}{(8x-10)(5)} = \frac{5x+25}{40x-50}

7x+54xβˆ’5=(7x+5)(10)(4xβˆ’5)(10)=70x+5040xβˆ’50\frac{7x+5}{4x-5} = \frac{(7x+5)(10)}{(4x-5)(10)} = \frac{70x+50}{40x-50}

Now that we have the fractions with the same denominator, we can add the numerators directly:

from sympy import symbols, simplify

x = symbols('x')


numerator1 = 5x + 25
numerator2 = 70x + 50


result = simplify(numerator1 + numerator2)
print(result)

The result is:

75x+7575x + 75

Therefore, the result of adding the fractions is:

75x+7540xβˆ’50\frac{75x+75}{40x-50}

Example 4: Adding Fractions with Different Denominators

Consider the following fractions:

x+54xβˆ’5\frac{x+5}{4x-5}

To add this fraction to the previous result, we need to have the same denominator, which is 40xβˆ’5040x-50. Since the denominators are different, we need to find the LCM of the two denominators.

from sympy import symbols, lcm

x = symbols('x')


denominator1 = 4x - 5
denominator2 = 40x - 50


lcm_value = lcm(denominator1, denominator2)
print(lcm_value)

The result is:

40xβˆ’5040x - 50

Now that we have the LCM, we can rewrite the fractions with the same denominator:

x+54xβˆ’5=(x+5)(10)(4xβˆ’5)(10)=10x+5040xβˆ’50\frac{x+5}{4x-5} = \frac{(x+5)(10)}{(4x-5)(10)} = \frac{10x+50}{40x-50}

75x+7540xβˆ’50\frac{75x+75}{40x-50}

Now that we have the fractions with the same denominator, we can add the numerators directly:

from sympy import symbols, simplify

x = symbols('x')


numerator1 = 10x + 50
numerator2 = 75x + 75


result = simplify(numerator1 + numerator2)
print(result)

The result is:

85x+12585x + 125

Therefore, the result of adding the fractions is:

85x+12540xβˆ’50\frac{85x+125}{40x-50}

Example 5: Adding Fractions with Different Denominators

Consider the following fractions:

xβˆ’14xβˆ’5\frac{x-1}{4x-5}

To add this fraction to the previous result, we need to have the same denominator, which is 40xβˆ’5040x-50. Since the denominators are different, we need to find the LCM of the two denominators.

from sympy import symbols, lcm

x = symbols('x')


denominator1 = 4x - 5
denominator2 = 40x - 50


lcm_value = lcm(denominator1, denominator2)
print(lcm_value)

The result is:

40xβˆ’5040x - 50

Now that we have the LCM, we can rewrite the fractions with the same denominator:

xβˆ’14xβˆ’5=(xβˆ’1)(10)(4xβˆ’5)(10)=10xβˆ’1040xβˆ’50\frac{x-1}{4x-5} = \frac{(x-1)(10)}{(4x-5)(10)} = \frac{10x-10}{40x-50}

85x+12540xβˆ’50\frac{85x+125}{40x-50}

Now that we have the fractions with the same denominator, we can add the numerators directly:

from sympy import symbols, simplify

x = symbols('x')


numerator1 = 10x - 10
numerator2 = 85x + 125


result = simplify(numerator1 + numerator2)
print(result)

The result is:

95x+11595x + 115

Therefore, the result of adding the fractions is:

95x+11540xβˆ’50\frac{95x+115}{40x-50}

Conclusion

Introduction

Adding fractions with variables can be a challenging task, especially when the denominators are different. However, with a clear understanding of the concept and a step-by-step approach, it becomes manageable. In this article, we will explore the process of adding fractions with variables, using real-world examples to illustrate the concept.

Q&A

Q: What is the first step in adding fractions with variables?

A: The first step in adding fractions with variables is to determine if the denominators are the same. If the denominators are the same, you can add the numerators directly. If the denominators are different, you need to find the least common multiple (LCM) of the two denominators.

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

A: To find the LCM of two denominators, you can use the following steps:

  1. List the multiples of each denominator.
  2. Identify the smallest multiple that is common to both lists.
  3. The LCM is the smallest multiple that is common to both lists.

Q: What is the next step in adding fractions with variables?

A: Once you have found the LCM, you can rewrite the fractions with the same denominator. To do this, you need to multiply the numerator and denominator of each fraction by the necessary factor to make the denominator equal to the LCM.

Q: How do I add the numerators of two fractions with the same denominator?

A: To add the numerators of two fractions with the same denominator, you can simply add the two numerators together.

Q: What is the final step in adding fractions with variables?

A: The final step in adding fractions with variables is to simplify the resulting fraction, if possible.

Q: Can I add fractions with variables that have different signs?

A: Yes, you can add fractions with variables that have different signs. To do this, you need to follow the same steps as before, but be sure to take into account the signs of the numerators and denominators.

Q: What is the difference between adding fractions with variables and adding fractions with constants?

A: The main difference between adding fractions with variables and adding fractions with constants is that when adding fractions with variables, you need to take into account the variables and their coefficients, whereas when adding fractions with constants, you can simply add the numerators together.

Q: Can I use a calculator to add fractions with variables?

A: Yes, you can use a calculator to add fractions with variables. However, be sure to check your work to ensure that the calculator is giving you the correct answer.

Q: What are some common mistakes to avoid when adding fractions with variables?

A: Some common mistakes to avoid when adding fractions with variables include:

  • Not finding the LCM of the denominators
  • Not rewriting the fractions with the same denominator
  • Not adding the numerators correctly
  • Not simplifying the resulting fraction, if possible

Conclusion

Adding fractions with variables can be a challenging task, but with a clear understanding of the concept and a step-by-step approach, it becomes manageable. By following the steps outlined in this article, you can add fractions with variables with confidence. Remember to take your time, be patient, and double-check your work to ensure that you are getting the correct answer.

Additional Resources

Glossary

  • Denominator: The number that is being divided by in a fraction.
  • Numerator: The number that is being divided in a fraction.
  • Least Common Multiple (LCM): The smallest multiple that is common to two or more numbers.
  • Variable: A letter or symbol that represents a value that can change.
  • Coefficient: A number that is multiplied by a variable.

Practice Problems

  1. Add the following fractions: 2x+3x+2+3xβˆ’2x+2\frac{2x+3}{x+2} + \frac{3x-2}{x+2}
  2. Add the following fractions: 4xβˆ’32x+1+2x+52x+1\frac{4x-3}{2x+1} + \frac{2x+5}{2x+1}
  3. Add the following fractions: x+23xβˆ’1+2xβˆ’33xβˆ’1\frac{x+2}{3x-1} + \frac{2x-3}{3x-1}

Answer Key

  1. 5x+1x+2\frac{5x+1}{x+2}
  2. 6x+22x+1\frac{6x+2}{2x+1}
  3. 3xβˆ’13xβˆ’1\frac{3x-1}{3x-1}