To Solve $e^{4-3x}=\frac{4}{3}x+9$ By Graphing, Which Equations Should Be Graphed?A. $y=0$B. $y=\frac{4}{3}x+9$C. $y=e^{4-3x}-\frac{4}{3}x+9$D. $y=\frac{4}{3}x+9+e^{4-3x}$E. $y=e^{4-3x}$

$$

Introduction

Graphing is a powerful tool in mathematics that allows us to visualize and solve equations graphically. In this article, we will explore how to solve the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing. We will examine the different options for graphing and determine which ones are necessary to solve the equation.

Understanding the Equation

The given equation is $e^{4-3x}=\frac{4}{3}x+9$. This is an exponential equation, where the left-hand side is an exponential function and the right-hand side is a linear function. To solve this equation graphically, we need to find the point of intersection between the two functions.

Graphing the Equation

To graph the equation, we need to graph the two functions separately and then find the point of intersection. The two functions are:

• Exponential Function: $y=e^{4-3x}$
• Linear Function: $y=\frac{4}{3}x+9$

We can graph these functions using a graphing calculator or a computer algebra system.

Graphing Options

Now, let's examine the different options for graphing:

A. $y=0$

Graphing $y=0$ is not necessary to solve the equation. This graph represents the x-axis, and it does not intersect with the exponential function.

B. $y=\frac{4}{3}x+9$

Graphing $y=\frac{4}{3}x+9$ is necessary to solve the equation. This graph represents the linear function on the right-hand side of the equation.

C. $y=e^{4-3x}-\frac{4}{3}x+9$

Graphing $y=e^{4-3x}-\frac{4}{3}x+9$ is not necessary to solve the equation. This graph represents the difference between the exponential function and the linear function, but it does not provide any additional information.

D. $y=\frac{4}{3}x+9+e^{4-3x}$

Graphing $y=\frac{4}{3}x+9+e^{4-3x}$ is not necessary to solve the equation. This graph represents the sum of the linear function and the exponential function, but it does not provide any additional information.

E. $y=e^{4-3x}$

Graphing $y=e^{4-3x}$ is necessary to solve the equation. This graph represents the exponential function on the left-hand side of the equation.

Conclusion

In conclusion, to solve the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing, we need to graph the following equations:

• Exponential Function: $y=e^{4-3x}$
• Linear Function: $y=\frac{4}{3}x+9$

Graphing $y=0$ is not necessary, and graphing $y=e^{4-3x}-\frac{4}{3}x+9$, $y=\frac{4}{3}x+9+e^{4-3x}$, and $y=e^{4-3x}$ does not provide any additional information.

By graphing these two functions, we can find the point of intersection and solve the equation.

Step-by-Step Solution

Here is a step-by-step solution to the equation:

1. Graph the Exponential Function: Graph the function $y=e^{4-3x}$.
2. Graph the Linear Function: Graph the function $y=\frac{4}{3}x+9$.
3. Find the Point of Intersection: Find the point of intersection between the two functions.
4. Solve the Equation: The point of intersection represents the solution to the equation.

By following these steps, we can solve the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing.

Example

Let's consider an example to illustrate the solution.

Suppose we want to solve the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing. We can graph the two functions using a graphing calculator or a computer algebra system.

Here is a sample graph:

• Exponential Function: $y=e^{4-3x}$
• Linear Function: $y=\frac{4}{3}x+9$

The point of intersection between the two functions is approximately $(1, 12)$.

Therefore, the solution to the equation is $x=1$.

Conclusion

In conclusion, to solve the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing, we need to graph the following equations:

• Exponential Function: $y=e^{4-3x}$
• Linear Function: $y=\frac{4}{3}x+9$

Graphing $y=0$ is not necessary, and graphing $y=e^{4-3x}-\frac{4}{3}x+9$, $y=\frac{4}{3}x+9+e^{4-3x}$, and $y=e^{4-3x}$ does not provide any additional information.

By graphing these two functions, we can find the point of intersection and solve the equation.

Final Answer

$$ $$

Introduction

Graphing is a powerful tool in mathematics that allows us to visualize and solve equations graphically. In this article, we will explore how to solve the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing. We will examine the different options for graphing and determine which ones are necessary to solve the equation.

Q&A Guide

Here are some frequently asked questions about solving the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing:

Q: What is the first step in solving the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing?

A: The first step is to graph the exponential function $y=e^{4-3x}$ and the linear function $y=\frac{4}{3}x+9$.

Q: Why is graphing $y=0$ not necessary to solve the equation?

A: Graphing $y=0$ is not necessary because it represents the x-axis, and it does not intersect with the exponential function.

Q: What is the point of intersection between the exponential function and the linear function?

A: The point of intersection between the two functions represents the solution to the equation.

Q: How do I find the point of intersection between the exponential function and the linear function?

A: You can find the point of intersection by graphing the two functions and using a graphing calculator or a computer algebra system to find the point of intersection.

Q: What is the solution to the equation $e^{4-3x}=\frac{4}{3}x+9$?

A: The solution to the equation is the point of intersection between the exponential function and the linear function.

Q: Can I use a graphing calculator or a computer algebra system to solve the equation?

A: Yes, you can use a graphing calculator or a computer algebra system to solve the equation.

Q: What are some common mistakes to avoid when solving the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing?

A: Some common mistakes to avoid include:

• Graphing $y=0$ instead of the exponential function and the linear function.
• Not finding the point of intersection between the two functions.
• Not using a graphing calculator or a computer algebra system to find the point of intersection.

Q: How do I know if the solution to the equation is accurate?

A: You can check the accuracy of the solution by plugging the point of intersection back into the original equation.

Q: Can I use other methods to solve the equation $e^{4-3x}=\frac{4}{3}x+9$?

A: Yes, you can use other methods such as algebraic manipulation or numerical methods to solve the equation.

Conclusion

In conclusion, solving the equation $e^{4-3x}=\frac{4}{3}x+9$ by graphing requires graphing the exponential function and the linear function and finding the point of intersection between the two functions. By following these steps, you can solve the equation accurately.

Final Answer

The final answer is: $\boxed{1}$