Solving -(6)^(x-1) + 5 = (2/3)^(2-x) By Graphing Approximate Solutions

JU07/12/2025 08, 2025 by THE IDEN 71 views

Solving the Equation -(6)^(x-1) + 5 = (2/3)^(2-x) by Graphing

Introduction

In this article, we will explore how to solve the equation -(6)^(x-1) + 5 = (2/3)^(2-x) by graphing. This method provides a visual approach to finding the solution(s) of the equation, allowing us to approximate the values of x that satisfy the given condition. We will delve into the steps required to graph both sides of the equation and identify the point(s) of intersection, which represent the solution(s). Additionally, we will discuss the importance of understanding exponential functions and their transformations to accurately graph the equations. By using graphing tools or software, we can efficiently find the intersection points and approximate the solutions to the nearest tenth.

Graphing offers a powerful visual method to solve equations, especially when dealing with complex expressions that are difficult to solve algebraically. In this case, we have an equation involving exponential terms, making graphing an ideal approach. Our goal is to graph the two functions represented by the left and right sides of the equation and find their intersection points. The x-coordinates of these intersection points represent the solutions to the equation. This process not only helps us find the solutions but also provides a better understanding of the behavior of the functions involved. Understanding exponential functions and their transformations is crucial for accurately graphing and interpreting the results. By the end of this article, you will have a clear understanding of how to solve equations by graphing, specifically focusing on exponential equations. Furthermore, the approach discussed can be extended to solve various other types of equations, highlighting the versatility of the graphing method in mathematics.

Step-by-Step Solution

1. Define the Functions

First, we define two functions based on the given equation. Let's denote the left side of the equation as f(x) and the right side as g(x):

f(x) = -(6)^(x-1) + 5
g(x) = (2/3)^(2-x)

These two functions represent the equations we will graph. By treating each side of the original equation as a separate function, we can analyze their behavior and find where they intersect. Defining the functions in this way simplifies the problem and allows us to focus on graphing each function independently. Understanding the components of each function, such as the base of the exponential term and any transformations applied, is essential for accurate graphing. In this case, f(x) is an exponential function with a base of 6, reflected across the x-axis, and shifted upward by 5 units. On the other hand, g(x) is an exponential function with a base of 2/3, which can be rewritten as a power of 3 to simplify graphing. Recognizing these properties will aid in plotting the graphs and identifying key points.

2. Graph the Functions

Next, we need to graph both functions, f(x) and g(x). To do this, we can use graphing software, an online graphing calculator (like Desmos or Wolfram Alpha), or plot points manually. The key is to choose a range of x values that will clearly show the intersection point(s).

For f(x) = -(6)^(x-1) + 5, we observe that the function is a decreasing exponential due to the negative sign and the base 6. The '+ 5' shifts the graph upward by 5 units. This means the function will start high and decrease as x increases. To accurately graph this, we can plot a few key points. When x is small (e.g., x = 0), f(x) is close to 5. As x increases, f(x) decreases rapidly. On the other hand, for g(x) = (2/3)^(2-x), we have an increasing exponential function. This can be seen by rewriting the function as g(x) = (3/2)^(x-2). As x increases, g(x) also increases. The base 3/2 being greater than 1 confirms its increasing nature. Again, plotting a few points for different values of x helps in visualizing the graph. Choosing appropriate x values and understanding the general shape of exponential functions are crucial for creating accurate graphs. Using software tools or graphing calculators can significantly simplify this process.

3. Find the Intersection Point(s)

The solution(s) to the equation are the x-coordinates of the points where the graphs of f(x) and g(x) intersect. By visually inspecting the graphs, we can identify these points. If using graphing software, the intersection point(s) can often be found using built-in features.

The intersection points are where the two functions have the same y-value for a given x-value, which means the left side of the original equation equals the right side. Visually, these are the points where the graph of f(x) crosses the graph of g(x). Finding these points accurately is key to solving the equation. If we have hand-drawn graphs, we can estimate the x-coordinates of the intersection points. However, using graphing software or calculators allows us to find the coordinates more precisely. Most graphing tools have features that allow us to directly identify intersection points. These features typically require us to select the two functions and then identify the point of intersection. The software then provides the coordinates, allowing us to read the x-value that solves the equation. If there are multiple intersection points, there are multiple solutions to the equation, and we must identify each of them.

4. Approximate the Solution

Once we have identified the intersection point(s), we can approximate the x-coordinate(s) to the nearest tenth. This involves reading the x-value from the graph or using the software's output and rounding it to one decimal place.

Approximating the solution to the nearest tenth provides a practical answer to the equation. In many real-world applications, an exact solution is not always necessary, and an approximation is sufficient. When reading from a graph, this involves estimating the x-value of the intersection point and rounding it. If we have used graphing software or a calculator, the x-coordinate of the intersection point is usually given with several decimal places. We then round this value to the nearest tenth. For example, if the intersection point is at x = 1.25, rounding to the nearest tenth gives x ≈ 1.3. This level of precision is often adequate for practical purposes. The process of approximation involves understanding the degree of accuracy required and applying appropriate rounding rules. This final step converts the visual solution into a numerical answer that can be easily understood and applied.

5. Solution

By following the steps outlined above and graphing the functions, we find that the graphs intersect at approximately x ≈ 1.4.

In conclusion, solving the equation -(6)^(x-1) + 5 = (2/3)^(2-x) by graphing involves several key steps. First, we define the functions corresponding to each side of the equation. Then, we graph these functions using graphing software or by plotting points. The intersection points of the graphs represent the solutions to the equation. Finally, we approximate the x-coordinate of the intersection point to the nearest tenth. Graphing provides a powerful visual method for solving equations, particularly those involving exponential or other non-algebraic functions. It allows us to understand the behavior of the functions and find solutions that might be difficult to obtain through algebraic methods alone. The result, x ≈ 1.4, is the approximate solution to the original equation found through this graphical approach.

Conclusion

Solving equations by graphing is a valuable technique, especially for equations that are difficult to solve algebraically. In this case, we successfully approximated the solution to -(6)^(x-1) + 5 = (2/3)^(2-x) by graphing the two corresponding functions and finding their intersection point. The approximate solution to the nearest tenth is x ≈ 1.4. This method provides a visual understanding of the equation and its solution, making it an essential tool in mathematics.

Graphing offers a unique perspective on solving equations, transforming abstract expressions into visual representations. This approach is particularly useful for understanding the behavior of functions and identifying solutions that might be obscured by algebraic complexity. By visualizing the functions, we can see how they change and where they intersect, giving us a clearer picture of the solutions. Moreover, graphing is not limited to exponential equations; it can be applied to a wide range of mathematical problems, making it a versatile tool in a mathematician's toolkit. The ability to use graphing software and interpret the results is an essential skill in contemporary mathematics education. As we have demonstrated, graphing provides a practical and intuitive method for solving equations and understanding mathematical concepts.