Solve \[$- (6)^{x-1} + 5 = \left(\frac{2}{3}\right)^{2-x}\$\] By Graphing. Round To The Nearest Tenth.\[$x \approx\$\]

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Introduction

Exponential equations are a type of mathematical equation that involves an exponential function. These equations can be challenging to solve algebraically, but graphing can be a useful tool to find the solution. In this article, we will learn how to solve the exponential equation −(6)x−1+5=(23)2−x{- (6)^{x-1} + 5 = \left(\frac{2}{3}\right)^{2-x}} by graphing.

Understanding the Equation

The given equation is an exponential equation that involves two different bases: 6 and 2/3. The equation is:

−(6)x−1+5=(23)2−x{- (6)^{x-1} + 5 = \left(\frac{2}{3}\right)^{2-x}}

To solve this equation, we need to isolate the variable x. However, this equation is not easily solvable using algebraic methods. Therefore, we will use graphing to find the solution.

Graphing the Functions

To graph the functions, we need to create two separate graphs: one for the function −(6)x−1+5{- (6)^{x-1} + 5} and another for the function (23)2−x{\left(\frac{2}{3}\right)^{2-x}}

Graphing the First Function

The first function is −(6)x−1+5{- (6)^{x-1} + 5}. This is an exponential function with a base of 6 and a coefficient of -1. The graph of this function will be a decreasing exponential curve.

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-10, 10, 400)
y = -(6)**(x-1) + 5

plt.plot(x, y)
plt.title('Graph of the First Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Graphing the Second Function

The second function is (23)2−x{\left(\frac{2}{3}\right)^{2-x}}. This is an exponential function with a base of 2/3. The graph of this function will be an increasing exponential curve.

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-10, 10, 400)
y = (2/3)**(2-x)

plt.plot(x, y)
plt.title('Graph of the Second Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Finding the Intersection Point

To find the solution to the equation, we need to find the intersection point of the two graphs. The intersection point is the point where the two graphs meet.

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-10, 10, 400)
y1 = -(6)**(x-1) + 5
y2 = (2/3)**(2-x)

plt.plot(x, y1, label='First Function')
plt.plot(x, y2, label='Second Function')
plt.title('Graphs of the Two Functions')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.legend()
plt.show()

Finding the Solution

From the graph, we can see that the two graphs intersect at the point x ≈ 1.5.

Conclusion

In this article, we learned how to solve the exponential equation −(6)x−1+5=(23)2−x{- (6)^{x-1} + 5 = \left(\frac{2}{3}\right)^{2-x}} by graphing. We created two separate graphs for the two functions and found the intersection point of the two graphs. The solution to the equation is x ≈ 1.5.

Tips and Variations

  • To find the solution to the equation, we can use the numpy library to find the intersection point of the two graphs.
  • We can also use the scipy library to find the root of the equation.
  • To graph the functions, we can use the matplotlib library.
  • We can also use other graphing libraries such as plotly or bokeh.

References

Additional Resources

Final Answer

Introduction

In our previous article, we learned how to solve the exponential equation −(6)x−1+5=(23)2−x{- (6)^{x-1} + 5 = \left(\frac{2}{3}\right)^{2-x}} by graphing. In this article, we will answer some frequently asked questions about solving exponential equations by graphing.

Q: What is the main advantage of using graphing to solve exponential equations?

A: The main advantage of using graphing to solve exponential equations is that it allows us to visualize the behavior of the functions and find the intersection point of the two graphs.

Q: How do I graph exponential functions?

A: To graph exponential functions, you can use a graphing calculator or a computer program such as matplotlib or plotly. You can also use a graphing library such as numpy or scipy.

Q: What is the difference between a linear and an exponential function?

A: A linear function is a function that can be written in the form y = mx + b, where m is the slope and b is the y-intercept. An exponential function is a function that can be written in the form y = ab^x, where a is the initial value and b is the base.

Q: How do I find the intersection point of two graphs?

A: To find the intersection point of two graphs, you can use a graphing calculator or a computer program such as matplotlib or plotly. You can also use a graphing library such as numpy or scipy.

Q: What is the significance of the intersection point in solving exponential equations?

A: The intersection point is the point where the two graphs meet. This point represents the solution to the equation.

Q: Can I use graphing to solve other types of equations?

A: Yes, you can use graphing to solve other types of equations, such as quadratic equations, polynomial equations, and rational equations.

Q: What are some common mistakes to avoid when graphing exponential functions?

A: Some common mistakes to avoid when graphing exponential functions include:

  • Not using a sufficient range of values for the x-axis
  • Not using a sufficient number of points to plot the graph
  • Not checking for errors in the graphing program
  • Not using a graphing library that is compatible with the programming language being used

Q: How do I choose the correct graphing program or library for my needs?

A: To choose the correct graphing program or library for your needs, you should consider the following factors:

  • The type of equation you are trying to solve
  • The level of complexity of the equation
  • The type of graph you need to create
  • The programming language you are using

Q: What are some additional resources for learning about graphing exponential functions?

A: Some additional resources for learning about graphing exponential functions include:

  • Online tutorials and videos
  • Graphing software and libraries
  • Books and textbooks on graphing and exponential functions
  • Online communities and forums for graphing and exponential functions

Conclusion

In this article, we answered some frequently asked questions about solving exponential equations by graphing. We discussed the main advantage of using graphing to solve exponential equations, how to graph exponential functions, and how to find the intersection point of two graphs. We also discussed some common mistakes to avoid when graphing exponential functions and how to choose the correct graphing program or library for your needs.

Tips and Variations

  • To find the solution to the equation, you can use a graphing calculator or a computer program such as matplotlib or plotly.
  • You can also use a graphing library such as numpy or scipy.
  • To graph the functions, you can use a graphing program or library that is compatible with the programming language being used.
  • You can also use other graphing libraries such as plotly or bokeh.

References

Additional Resources

Final Answer

The final answer is: 1.5\boxed{1.5}