Solve: [tex]9^{2 x+1}=9^{3 x-2}[/tex]
Answer (2)
Set the exponents equal to each other: 2 x + 1 = 3 x − 2 .
Rearrange the equation to isolate x: 3 x − 2 x = 1 + 2 .
Simplify the equation: x = 3 .
Verify the solution by substituting x back into the original equation: 9 2 ( 3 ) + 1 = 9 3 ( 3 ) − 2 .
The final answer is 3 .
Explanation
Equating the exponents We are given the equation 9 2 x + 1 = 9 3 x − 2 . Since the bases are equal, we can set the exponents equal to each other. This gives us the equation 2 x + 1 = 3 x − 2 .
Isolating x Now, we need to solve for x . Let's rearrange the equation to isolate x on one side. Subtract 2 x from both sides: 2 x + 1 − 2 x = 3 x − 2 − 2 x , which simplifies to 1 = x − 2 .
Solving for x Next, add 2 to both sides of the equation: 1 + 2 = x − 2 + 2 , which simplifies to 3 = x . Therefore, x = 3 .
Verification To check our answer, substitute x = 3 back into the original equation: 9 2 ( 3 ) + 1 = 9 3 ( 3 ) − 2 . This simplifies to 9 6 + 1 = 9 9 − 2 , which is 9 7 = 9 7 . Since this is true, our solution is correct.
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, they are used to model compound interest. Imagine you invest 1000 inana cco u n tt ha tp a ys 5 A = 1000(1.05)^t$. Solving exponential equations helps you determine how long it will take for your investment to reach a certain value. Similarly, in physics, exponential decay models the decrease in the amount of a radioactive substance over time.
To solve the equation 9 2 x + 1 = 9 3 x − 2 , we set the exponents equal to each other resulting in x = 3 . Verification shows both sides of the equation equal when substituted back in, confirming our solution. Therefore, the final answer is x = 3 .
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