Solve For $x$ In The Equation $t {4x-3}=t {3x}$.Type Your Answer

Mar 8, 2025 by ADMIN 69 views

Solving for x in the Equation t^(4x-3) = t^(3x)

Introduction

In this article, we will delve into solving for x in the given equation t^(4x-3) = t^(3x). This equation involves exponentiation and can be solved using algebraic techniques. We will break down the solution step by step, providing a clear and concise explanation of each step.

Understanding the Equation

The given equation is t^(4x-3) = t^(3x). This equation involves two exponential expressions with the same base, t. The exponents are 4x-3 and 3x, respectively. To solve for x, we need to equate the exponents and solve for x.

Equating the Exponents

Since the bases are the same, we can equate the exponents:

4x - 3 = 3x

Solving for x

To solve for x, we need to isolate x on one side of the equation. We can do this by adding 3 to both sides of the equation:

4x - 3 + 3 = 3x + 3

This simplifies to:

4x = 3x + 3

Next, we can subtract 3x from both sides of the equation:

4x - 3x = 3x - 3x + 3

This simplifies to:

x = 3

Verifying the Solution

To verify the solution, we can substitute x = 3 back into the original equation:

t^(4(3)-3) = t^(3(3))

This simplifies to:

t^(9) = t^(9)

Since the bases are the same and the exponents are equal, the equation is true. Therefore, x = 3 is a valid solution.

Conclusion

In this article, we solved for x in the equation t^(4x-3) = t^(3x). We equated the exponents and solved for x using algebraic techniques. The solution is x = 3, which can be verified by substituting x back into the original equation.

Frequently Asked Questions

Q: What is the base of the exponential expressions in the equation? A: The base of the exponential expressions is t.
Q: How do we solve for x in the equation? A: We equate the exponents and solve for x using algebraic techniques.
Q: Is x = 3 a valid solution? A: Yes, x = 3 is a valid solution, which can be verified by substituting x back into the original equation.

Additional Resources

For more information on solving exponential equations, see [1].
For more information on algebraic techniques, see [2].

References

[1] Khan Academy. (n.d.). Solving Exponential Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f0f21d/solving-exponential-equations/v/solving-exponential-equations [2] Mathway. (n.d.). Algebra. Retrieved from https://www.mathway.com/subjects/algebra

Step-by-Step Solution

Step 1: Equate the Exponents

4x - 3 = 3x

Step 2: Add 3 to Both Sides

4x - 3 + 3 = 3x + 3

Step 3: Simplify

4x = 3x + 3

Step 4: Subtract 3x from Both Sides

4x - 3x = 3x - 3x + 3

Step 5: Simplify

x = 3

Step 6: Verify the Solution

t^(4(3)-3) = t^(3(3))

t^(9) = t^(9)

Step 7: Conclusion

x = 3 is a valid solution.

Introduction

In our previous article, we solved for x in the equation t^(4x-3) = t^(3x). We equated the exponents and solved for x using algebraic techniques. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q&A

Q: What is the base of the exponential expressions in the equation?

A: The base of the exponential expressions is t.

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

A: We equate the exponents and solve for x using algebraic techniques.

Q: Is x = 3 a valid solution?

A: Yes, x = 3 is a valid solution, which can be verified by substituting x back into the original equation.

Q: What if the equation has a different base?

A: If the equation has a different base, we can still solve for x by equating the exponents. However, we need to make sure that the bases are the same before equating the exponents.

Q: Can we use logarithms to solve for x?

A: Yes, we can use logarithms to solve for x. By taking the logarithm of both sides of the equation, we can eliminate the exponent and solve for x.

Q: What if the equation has a negative exponent?

A: If the equation has a negative exponent, we can still solve for x by equating the exponents. However, we need to make sure that the bases are the same before equating the exponents.

Q: Can we use algebraic techniques to solve for x in other types of equations?

A: Yes, we can use algebraic techniques to solve for x in other types of equations. However, we need to make sure that the equation is in a form that can be solved using algebraic techniques.

Q: What if I get stuck while solving for x?

A: If you get stuck while solving for x, you can try breaking down the problem into smaller steps or seeking help from a teacher or tutor.

Tips and Tricks

Make sure to read the problem carefully and understand what is being asked.
Use algebraic techniques to solve for x.
Check your work by substituting x back into the original equation.
Use logarithms to solve for x if the equation has a negative exponent.
Break down the problem into smaller steps if you get stuck.

Common Mistakes

Forgetting to equate the exponents.
Not checking the work by substituting x back into the original equation.
Using the wrong algebraic technique.
Not using logarithms when the equation has a negative exponent.

Conclusion

In this article, we provided a Q&A section to address any questions or concerns that readers may have. We also provided tips and tricks for solving for x in the equation t^(4x-3) = t^(3x). By following these tips and tricks, you can avoid common mistakes and solve for x with confidence.

Frequently Asked Questions

Q: What is the base of the exponential expressions in the equation? A: The base of the exponential expressions is t.
Q: How do we solve for x in the equation? A: We equate the exponents and solve for x using algebraic techniques.
Q: Is x = 3 a valid solution? A: Yes, x = 3 is a valid solution, which can be verified by substituting x back into the original equation.

Additional Resources

For more information on solving exponential equations, see [1].
For more information on algebraic techniques, see [2].
For more information on logarithms, see [3].