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To eliminate the parameter \( t \) from the given parametric equations:

\[
x = e^{-7t}, \quad y = e^{7t}
\]

we can start by solving each equation for \( t \) and then equating them.

### Step-by-Step Solution:

1. **Solve for \( t \) from each equation:**

   - For \( x = e^{-7t} \):
     \[
     \ln(x) = -7t \Rightarrow t = -\frac{\ln(x)}{7}
     \]

   - For \( y = e^{7t} \):
     \[
     \ln(y) = 7t \Rightarrow t = \frac{\ln(y)}{7}
     \]

2. **Equate the expressions for \( t \):**

   Since both expressions represent \( t \), we can set them equal to each other:
   \[
   -\frac{\ln(x)}{7} = \frac{\ln(y)}{7}
   \]

3. **Simplify the equation:**

   Multiply both sides by 7 to eliminate the denominator:
   \[
   -\ln(x) = \ln(y)
   \]

4. **Rewrite in terms of \( x \) and \( y \):**

   This simplifies to:
   \[
   \ln(y) = -\ln(x)
   \]

5. **Combine logarithmic terms:**

   Using the property \(\ln(a) = -\ln(b) \Rightarrow a = \frac{1}{b}\), we get:
   \[
   y = \frac{1}{x}
   \]

### Final Answer:

The Cartesian equation of the curve is:
\[
y = \frac{1}{x}
\]

Would you like further details on this solution or have additional questions?

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Here are 5 related questions for further exploration:

1. How would you graph the curve \( y = \frac{1}{x} \) in the Cartesian plane?
2. What happens to the curve as \( x \) approaches zero?
3. How would this solution change if the powers in \( e^{-7t} \) and \( e^{7t} \) were different?
4. What is the domain and range of the curve \( y = \frac{1}{x} \)?
5. How could we verify this solution using another method?

**Tip:** When dealing with exponential equations, logarithms are often helpful for isolating variables or eliminating parameters.

Consider the following:

x = e^(-7t), y = e^(7t)

(a) Eliminate the parameter to find a Cartesian equation of the curve.

Explore a step-by-step solution for eliminating the parameter t in the exponential equations x = e^(-7t) and y = e^(7t) to find a Cartesian equation. This problem covers algebraic manipulation, logarithmic properties, and parametric equations. Suitable for high school students in Grades 10-12.

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