Solving For X In The Exponential Equation (2/3)^(x+5) = (9/4)^(x-1)

In mathematics, exponential equations play a crucial role in modeling various real-world phenomena, from population growth to radioactive decay. Solving these equations often requires a solid understanding of exponent rules and algebraic manipulation. In this article, we will delve into the step-by-step solution of the exponential equation $\\left(\\frac{2}{3}\\right)^{{x+5}=\\left(\\frac{9}{4}\\right)}{x-1}$, demonstrating the techniques used to isolate the variable x and determine its value. Our focus will be on providing a clear and comprehensive explanation, ensuring that readers can grasp the underlying concepts and apply them to similar problems.

First, let's understand the equation we are working with: $\\left(\\frac{2}{3}\\right)^{{x+5}=\\left(\\frac{9}{4}\\right)}{x-1}$. The key to solving this equation lies in recognizing that both sides can be expressed with the same base. Notice that $\\frac{9}{4}$$ is the reciprocal of $\\frac{4}{9}$$, which is the square of $\\frac{2}{3}$$. By expressing both sides of the equation with the base $\\frac{2}{3}$$, we can equate the exponents and solve for x. This approach simplifies the equation and allows us to use the properties of exponents to our advantage. Understanding the relationship between the bases is the first critical step in unraveling this exponential equation. The ability to manipulate bases and exponents is a fundamental skill in solving exponential equations, and mastering this technique will greatly enhance your problem-solving capabilities in mathematics.

To proceed, we rewrite $\\left(\\frac{9}{4}\\right)$$ as $\\left(\\frac{3}{2}\\right)^2$$. Then, we rewrite $\\left(\\frac{3}{2}\\right)$$ as $\\left(\\frac{2}{3}\\right)^{-1}$$. This transformation is crucial because it allows us to express both sides of the equation with the same base, which is $\\frac{2}{3}$$. When we have the same base on both sides of an exponential equation, we can equate the exponents and solve for the variable. By rewriting the equation in this form, we set the stage for a straightforward algebraic solution. This step highlights the importance of recognizing reciprocal relationships and utilizing negative exponents to manipulate expressions. Once both sides of the equation share a common base, the problem simplifies significantly, making it easier to isolate and solve for x. The manipulation of bases and exponents is a core skill in dealing with exponential equations, and this step exemplifies its practical application. Now we can express the right side of the equation as $\\left[\\left(\\frac{2}{3}\\right)^{-1}\\right]{2(x-1)}$$, which simplifies to $\\left(\\frac{2}{3}\\right)^{-2(x-1)}$$. With this transformation, the equation now reads $\\left(\\frac{2}{3}\\right)^{x+5} = \\left(\\frac{2}{3}\\right)^{-2(x-1)}$$. The equation is now in a form where we can directly compare the exponents.

Now that we have expressed both sides of the equation with the same base, we can equate the exponents. This is a fundamental property of exponential equations: if $a^m = a^n$, then m = n. Applying this property to our equation, we set the exponent on the left side equal to the exponent on the right side, resulting in the equation $x + 5 = -2(x - 1)$. This linear equation is much simpler to solve than the original exponential equation. The transformation from exponential form to a linear equation is a key step in finding the value of x. From here, we can use basic algebraic techniques to isolate x and determine its value. The ability to transition from exponential equations to linear equations is a powerful tool in mathematical problem-solving. Once we have a linear equation, the solution process becomes more straightforward, involving steps such as distribution, combining like terms, and isolating the variable. Equating exponents is a critical technique in solving exponential equations, and this step demonstrates its practical application in simplifying a complex problem. Now, let’s proceed to solve this linear equation for x. By carefully following the steps of algebraic manipulation, we will arrive at the final solution.

To solve the linear equation $x + 5 = -2(x - 1)$, we first distribute the -2 on the right side of the equation. This gives us $x + 5 = -2x + 2$. Distributing the constant is a crucial step in simplifying the equation and allows us to combine like terms. This step is based on the distributive property of multiplication over addition, which states that a(b + c) = ab + ac. In our case, we are distributing -2 over the expression (x - 1), which involves multiplying -2 by both x and -1. Accurate distribution is essential for maintaining the equality of the equation and avoiding errors in the solution process. Once the distribution is complete, the equation becomes easier to manipulate and solve for x. The distributive property is a fundamental concept in algebra, and its application is essential in solving a wide range of equations. By correctly applying the distributive property, we can transform complex expressions into simpler forms, making the solution process more manageable. Now that we have distributed the -2, we can proceed to combine like terms and isolate x.

Next, we want to get all the x terms on one side of the equation and the constants on the other side. To do this, we can add 2x to both sides of the equation, which gives us $3x + 5 = 2$. Adding the same term to both sides of the equation maintains the equality and helps to isolate the variable. In this case, adding 2x to both sides eliminates the x term from the right side and combines it with the x term on the left side. This is a common strategy in solving linear equations, as it simplifies the equation by grouping similar terms together. By strategically adding or subtracting terms from both sides, we can gradually isolate the variable and make the equation easier to solve. The goal is to manipulate the equation in such a way that x is by itself on one side, allowing us to determine its value. Adding 2x to both sides is a key step in achieving this goal. Now that we have combined the x terms, we can move on to isolating x further by dealing with the constant term.

Now, we subtract 5 from both sides of the equation to isolate the term with x. This yields $3x = 2 - 5$, which simplifies to $3x = -3$. Subtracting the same constant from both sides of the equation maintains the equality and helps to further isolate the variable. In this case, subtracting 5 from both sides eliminates the constant term from the left side, leaving only the term with x. This is another common strategy in solving linear equations, as it gradually brings us closer to the solution. By performing inverse operations on both sides of the equation, we can systematically eliminate terms and isolate the variable. The goal is to manipulate the equation until x is by itself on one side, allowing us to determine its value. Subtracting 5 from both sides is a crucial step in achieving this goal. With the x term now isolated on one side, we can move on to the final step of dividing by the coefficient of x to find the value of x.

Finally, we divide both sides of the equation by 3 to solve for x. This gives us $x = \\frac{-3}{3}$$, which simplifies to $x = -1$. Dividing both sides of the equation by the same non-zero number maintains the equality and isolates the variable. In this case, dividing by 3, the coefficient of x, gives us the value of x. This is the final step in solving the linear equation, and it provides the solution to the original exponential equation. By performing this division, we have successfully determined the value of x that satisfies the equation. The process of solving linear equations involves a series of steps, including distribution, combining like terms, and performing inverse operations to isolate the variable. Each step is carefully designed to maintain the equality of the equation and bring us closer to the solution. The ability to solve linear equations is a fundamental skill in mathematics, and it is essential for solving a wide range of problems. Therefore, the solution to the exponential equation $\\left(\\frac{2}{3}\\right)^{{x+5}=\\left(\\frac{9}{4}\\right)}{x-1}$$ is $x = -1$.

To ensure the correctness of our solution, it is always a good practice to verify the solution by substituting the value of x back into the original equation. This process helps to catch any potential errors made during the solving process and confirms that the solution satisfies the equation. In our case, we found that $x = -1$$, so we will substitute this value into the original equation $\\left(\\frac{2}{3}\\right)^{{x+5}=\\left(\\frac{9}{4}\\right)}{x-1}$$. If both sides of the equation are equal after the substitution, then our solution is correct. Verification is an important step in problem-solving, as it provides confidence in the accuracy of the solution. It also helps to reinforce the understanding of the concepts and techniques used to solve the problem. By substituting the solution back into the original equation, we are essentially reversing the steps we took to solve it, ensuring that the equation holds true. This process can also help to identify any mistakes made during the solution process, allowing for correction and a more thorough understanding of the problem. Now, let's substitute $x = -1$$ into the original equation and see if it holds true.

Substituting $x = -1$$ into the left side of the equation, we get $\\left(\\frac{2}{3}\\right)^{-1+5} = \\left(\\frac{2}{3}\\right)^{4}$$. Calculating this, we have $\\left(\\frac{2}{3}\\right)^{4} = \\frac{2^4}{34} = \\frac{16}{81}$$. The left side of the equation simplifies to $\\frac{16}{81}$$ when $x = -1$$. This calculation involves raising a fraction to a power, which means raising both the numerator and the denominator to that power. The ability to accurately calculate exponents and fractions is essential for verifying solutions in exponential equations. By substituting the value of x into the left side of the equation and simplifying, we obtain a numerical value that we can compare to the value obtained by substituting x into the right side of the equation. This comparison is the key to verifying whether our solution is correct. If the left and right sides of the equation evaluate to the same value, then we can be confident that our solution is accurate. Now, let's substitute $x = -1$$ into the right side of the equation and see if we get the same value.

Substituting $x = -1$$ into the right side of the equation, we get $\\left(\\frac{9}{4}\\right)^{-1-1} = \\left(\\frac{9}{4}\\right)^{-2}$$. Recall that a negative exponent means we take the reciprocal of the base raised to the positive exponent. So, $\\left(\\frac{9}{4}\\right)^{-2} = \\left(\\frac{4}{9}\\right)^{2}$$. Calculating this, we have $\\left(\\frac{4}{9}\\right)^{2} = \\frac{4^2}{92} = \\frac{16}{81}$$. The right side of the equation also simplifies to $\\frac{16}{81}$$ when $x = -1$$. This calculation involves understanding and applying the properties of negative exponents, as well as the rules for raising fractions to powers. The ability to manipulate exponents and fractions is crucial for verifying solutions in exponential equations. By substituting the value of x into the right side of the equation and simplifying, we obtain a numerical value that we can compare to the value we obtained from the left side. If the two values are equal, then we have successfully verified our solution. In this case, both sides of the equation evaluate to $\\frac{16}{81}$$, which confirms that our solution is correct.

Since both sides of the equation equal $\\frac{16}{81}$$ when $x = -1$$, our solution is verified. This confirms that the value $x = -1$$ is the correct solution to the exponential equation $\\left(\\frac{2}{3}\\right)^{{x+5}=\\left(\\frac{9}{4}\\right)}{x-1}$$. Verification is a crucial step in the problem-solving process, as it provides assurance that the solution obtained is accurate and satisfies the given equation. By substituting the solution back into the original equation and showing that both sides are equal, we have demonstrated the correctness of our answer. This process also reinforces our understanding of the concepts and techniques used to solve the problem. The ability to verify solutions is a valuable skill in mathematics, as it helps to prevent errors and build confidence in our problem-solving abilities. In this case, the verification process has confirmed that our solution is indeed correct.

In conclusion, we have successfully solved the exponential equation $\\left(\\frac{2}{3}\\right)^{{x+5}=\\left(\\frac{9}{4}\\right)}{x-1}$$ and found that $x = -1$$. The solution process involved several key steps, including recognizing the relationship between the bases, rewriting the equation with a common base, equating the exponents, solving the resulting linear equation, and verifying the solution. Each of these steps is essential for solving exponential equations effectively. The ability to manipulate exponents and fractions, as well as solve linear equations, is crucial for success in this area of mathematics. By understanding the properties of exponents and applying them strategically, we can simplify complex equations and find solutions. The verification process is also an important part of the solution, as it ensures the accuracy of our answer. In this case, we have demonstrated the step-by-step solution of the given exponential equation, providing a clear and comprehensive explanation of the techniques used. This example serves as a valuable illustration of how to approach and solve similar problems in mathematics. Mastering these techniques will enhance your problem-solving skills and deepen your understanding of exponential equations.

• Understanding Exponential Equations: Recognize the fundamental properties of exponents and how they apply to solving equations.
• Base Manipulation: Learn to rewrite expressions with a common base to simplify equations.
• Equating Exponents: Understand that if $a^m = a^n$, then m = n, and apply this principle to solve equations.
• Solving Linear Equations: Master the techniques for solving linear equations, including distribution, combining like terms, and isolating the variable.
• Verification: Always verify your solution by substituting it back into the original equation.

By following these key takeaways, you can effectively tackle a wide range of exponential equations and enhance your problem-solving abilities in mathematics. The principles and techniques discussed in this article provide a solid foundation for further exploration of exponential functions and their applications in various fields.