The Equation and the Point P

Oct 13, 2025 by windailys.com 128 views

Introduction

In this article, we will explore the concept of solving for the value of a variable in a given equation. We will use a specific example to demonstrate the steps involved in solving for the value of a. The example involves a point P that lies on the graph of the equation x - y = 16. We will use algebraic manipulations to solve for the value of a.

The Equation and the Point P

The equation given is x - y = 16. We are also given a point P with coordinates $\left(\frac{a^3+1}{a^3}, \frac{a^3-1}{a^3}\right)$ . The point P lies on the graph of the equation x - y = 16.

Substituting the Coordinates of P into the Equation

To find the value of a, we can substitute the coordinates of P into the equation x - y = 16. We get:

$\frac{a^3+1}{a^3} - \frac{a^3-1}{a^3} = 16$

Simplifying the Equation

We can simplify the equation by finding a common denominator:

$\frac{(a^3+1) - (a^3-1)}{a^3} = 16$

$\frac{2}{a^3} = 16$

Multiplying Both Sides by $a^3$

To get rid of the fraction, we can multiply both sides of the equation by $a^3$ :

$2 = 16a^3$

Dividing Both Sides by 16

To isolate $a^3$ , we can divide both sides of the equation by 16:

$\frac{2}{16} = a^3$

$\frac{1}{8} = a^3$

Taking the Cube Root of Both Sides

To find the value of a, we can take the cube root of both sides of the equation:

$a = \sqrt[3]{\frac{1}{8}}$

Simplifying the Cube Root

We can simplify the cube root by recognizing that $\frac{1}{8}$ is equal to $(\frac{1}{2})^3$ :

$a = \sqrt[3]{(\frac{1}{2})^3}$

$a = \frac{1}{2}$

Conclusion

In this article, we used algebraic manipulations to solve for the value of a in a given equation. We substituted the coordinates of a point P into the equation x - y = 16 and simplified the resulting equation to find the value of a. The final answer is $\frac{1}{2}$ .

Final Answer

The final answer is $\boxed{\frac{1}{2}}$ .

Additional Information

The equation x - y = 16 is a linear equation in two variables. The point P with coordinates $\left(\frac{a^3+1}{a^3}, \frac{a^3-1}{a^3}\right)$ lies on the graph of the equation. The value of a can be found by substituting the coordinates of P into the equation and solving for a.

References

[1] Algebra, 2nd edition, Michael Artin, Prentice Hall, 2010.
[2] Calculus, 3rd edition, Michael Spivak, Publish or Perish, 2008.

Introduction
The Equation and the Point P
Substituting the Coordinates of P into the Equation
Simplifying the Equation
Multiplying Both Sides by $a^3$
Dividing Both Sides by 16
Taking the Cube Root of Both Sides
Simplifying the Cube Root
Conclusion
Final Answer
Additional Information
References
Table of Contents
Q&A: Solving for the Value of a in a Given Equation =====================================================

Introduction

In our previous article, we explored the concept of solving for the value of a variable in a given equation. We used a specific example to demonstrate the steps involved in solving for the value of a. In this article, we will answer some frequently asked questions related to solving for the value of a in a given equation.

Q: What is the first step in solving for the value of a in a given equation?

A: The first step in solving for the value of a in a given equation is to substitute the coordinates of the point P into the equation. This will give us an equation that we can simplify and solve for the value of a.

Q: How do I simplify the equation after substituting the coordinates of P?

A: To simplify the equation, we can use algebraic manipulations such as finding a common denominator, combining like terms, and isolating the variable a.

Q: What is the significance of the cube root in solving for the value of a?

A: The cube root is used to find the value of a when the equation is in the form of $a^3 = \frac{1}{8}$ . Taking the cube root of both sides of the equation gives us the value of a.

Q: Can I use other methods to solve for the value of a?

A: Yes, there are other methods to solve for the value of a, such as using numerical methods or graphing the equation. However, the method we used in this article is a straightforward and algebraic approach.

Q: What are some common mistakes to avoid when solving for the value of a?

A: Some common mistakes to avoid when solving for the value of a include:

Not substituting the coordinates of P into the equation
Not simplifying the equation properly
Not isolating the variable a
Not checking the solution for extraneous solutions

Q: How do I check if my solution is correct?

A: To check if your solution is correct, you can substitute the value of a back into the original equation and verify that it is true. You can also use a calculator or graphing software to check the solution.

Q: Can I apply this method to other types of equations?

A: Yes, the method we used in this article can be applied to other types of equations, such as quadratic equations or polynomial equations. However, the specific steps may vary depending on the type of equation.

Conclusion

In this article, we answered some frequently asked questions related to solving for the value of a in a given equation. We provided step-by-step instructions and examples to help you understand the concept. Remember to always check your solution for extraneous solutions and to use algebraic manipulations to simplify the equation.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$ .

Additional Information

[1] Algebra, 2nd edition, Michael Artin, Prentice Hall, 2010.
[2] Calculus, 3rd edition, Michael Spivak, Publish or Perish, 2008.

Introduction
Q: What is the first step in solving for the value of a in a given equation?
Q: How do I simplify the equation after substituting the coordinates of P?
Q: What is the significance of the cube root in solving for the value of a?
Q: Can I use other methods to solve for the value of a?
Q: What are some common mistakes to avoid when solving for the value of a?
Q: How do I check if my solution is correct?
Q: Can I apply this method to other types of equations?
Conclusion
Final Answer
Additional Information
Table of Contents