Solve The Equation:${ 65x - \frac{x}{15} = 65 }$

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Introduction

In this article, we will delve into the world of algebra and solve a linear equation that involves fractions. The equation we will be working with is 65xβˆ’x15=6565x - \frac{x}{15} = 65. Our goal is to isolate the variable xx and find its value. We will use various algebraic techniques to simplify the equation and solve for xx.

Understanding the Equation

The given equation is 65xβˆ’x15=6565x - \frac{x}{15} = 65. This equation involves a linear term 65x65x and a fractional term βˆ’x15-\frac{x}{15}. To solve for xx, we need to get rid of the fraction and combine like terms.

Step 1: Get Rid of the Fraction

To eliminate the fraction, we can multiply both sides of the equation by the denominator of the fraction, which is 1515. This will allow us to work with whole numbers and simplify the equation.

15(65x - \frac{x}{15}) = 15(65)

Simplifying the Equation

After multiplying both sides by 1515, we get:

975x - x = 975

Combining Like Terms

Now, we can combine the like terms on the left-hand side of the equation:

974x = 975

Solving for x

To solve for xx, we need to isolate the variable. We can do this by dividing both sides of the equation by 974974:

x = \frac{975}{974}

Simplifying the Fraction

The fraction 975974\frac{975}{974} can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 11. Therefore, the simplified fraction is:

x = \frac{975}{974}

Conclusion

In this article, we solved the equation 65xβˆ’x15=6565x - \frac{x}{15} = 65 using algebraic techniques. We started by getting rid of the fraction, then combined like terms, and finally solved for xx. The solution to the equation is x=975974x = \frac{975}{974}.

Final Answer

The final answer to the equation 65xβˆ’x15=6565x - \frac{x}{15} = 65 is:

x = \frac{975}{974}

Related Topics

  • Solving linear equations
  • Simplifying fractions
  • Algebraic techniques

Glossary

  • Linear equation: An equation in which the highest power of the variable is 11.
  • Fraction: A way of expressing a part of a whole as a ratio of two numbers.
  • Algebraic technique: A method used to solve equations and manipulate expressions.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Linear Algebra" by Jim Hefferon
  • [3] "Solving Equations" by Math Open Reference

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Introduction

In our previous article, we solved the equation 65xβˆ’x15=6565x - \frac{x}{15} = 65 using algebraic techniques. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is the equation 65xβˆ’x15=6565x - \frac{x}{15} = 65?

A: The equation 65xβˆ’x15=6565x - \frac{x}{15} = 65 is a linear equation that involves a fraction. It can be solved using algebraic techniques to find the value of the variable xx.

Q: How do I get rid of the fraction in the equation?

A: To eliminate the fraction, you can multiply both sides of the equation by the denominator of the fraction, which is 1515. This will allow you to work with whole numbers and simplify the equation.

Q: What is the simplified equation after multiplying by 15?

A: After multiplying both sides by 1515, the simplified equation is:

975x - x = 975

Q: How do I combine like terms in the equation?

A: To combine like terms, you can add or subtract the coefficients of the terms with the same variable. In this case, you can combine the 975x975x and βˆ’x-x terms to get:

974x = 975

Q: How do I solve for x in the equation?

A: To solve for xx, you need to isolate the variable. You can do this by dividing both sides of the equation by 974974:

x = \frac{975}{974}

Q: Can I simplify the fraction 975974\frac{975}{974}?

A: Yes, the fraction 975974\frac{975}{974} can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 11. Therefore, the simplified fraction is:

x = \frac{975}{974}

Q: What is the final answer to the equation 65xβˆ’x15=6565x - \frac{x}{15} = 65?

A: The final answer to the equation 65xβˆ’x15=6565x - \frac{x}{15} = 65 is:

x = \frac{975}{974}

Q: What are some related topics to solving linear equations?

A: Some related topics to solving linear equations include:

  • Simplifying fractions
  • Algebraic techniques
  • Solving quadratic equations

Q: Where can I find more information on solving linear equations?

A: You can find more information on solving linear equations in textbooks, online resources, and math websites. Some recommended resources include:

  • "Algebra" by Michael Artin
  • "Linear Algebra" by Jim Hefferon
  • "Solving Equations" by Math Open Reference

Conclusion

In this Q&A guide, we provided answers to common questions about solving the equation 65xβˆ’x15=6565x - \frac{x}{15} = 65. We hope this guide has helped you understand the solution and answer any questions you may have.