https://r2.erweima.ai/mathbot/image/17253396860533djhszbt.jpg

The image provides a system of two equations:

\[
\begin{cases}
\frac{\pi}{x \sqrt{1 - y^2}} = 2 \\
e^{-\frac{y\pi}{\sqrt{1 - y^2} \cdot \pi}} = 0.417
\end{cases}
\]

We need to find the values of \(x\) and \(y\) that satisfy both equations.

### Step 1: Solving the First Equation

Given:

\[
\frac{\pi}{x \sqrt{1 - y^2}} = 2
\]

Rearrange to solve for \(x\):

\[
x \sqrt{1 - y^2} = \frac{\pi}{2}
\]

\[
x = \frac{\pi}{2 \sqrt{1 - y^2}}
\]

### Step 2: Solving the Second Equation

Given:

\[
e^{-\frac{y\pi}{\sqrt{1 - y^2} \cdot \pi}} = 0.417
\]

Simplify the exponent:

\[
-\frac{y\pi}{\sqrt{1 - y^2} \cdot \pi} = -\frac{y}{\sqrt{1 - y^2}}
\]

Let:

\[
e^{-\frac{y}{\sqrt{1 - y^2}}} = 0.417
\]

Take the natural logarithm of both sides:

\[
-\frac{y}{\sqrt{1 - y^2}} = \ln(0.417)
\]

### Step 3: Solving the System

From the logarithm calculation, we have:

\[
-\frac{y}{\sqrt{1 - y^2}} \approx \ln(0.417) \approx -0.874
\]

\[
\frac{y}{\sqrt{1 - y^2}} = 0.874
\]

Square both sides:

\[
\frac{y^2}{1 - y^2} = 0.764
\]

\[
y^2 = 0.764 - 0.764y^2
\]

\[
1.764y^2 = 0.764
\]

\[
y^2 = \frac{0.764}{1.764}
\]

\[
y^2 \approx 0.433
\]

\[
y \approx \pm \sqrt{0.433} \approx \pm 0.658
\]

### Step 4: Finding \( x \)

Substituting \( y \approx 0.658 \) into the expression for \( x \):

\[
x = \frac{\pi}{2 \sqrt{1 - (0.658)^2}}
\]

\[
x = \frac{\pi}{2 \sqrt{1 - 0.433}}
\]

\[
x = \frac{\pi}{2 \sqrt{0.567}}
\]

\[
x \approx \frac{3.14}{2 \times 0.753} \approx 2.084
\]

Thus, approximate solutions are:

\[
x \approx 2.084 \quad \text{and} \quad y \approx \pm 0.658
\]

Would you like a detailed explanation, or do you have any questions?

### Related Questions

1. How do we determine the validity of solutions in systems involving trigonometric identities?
2. What are common methods to solve non-linear equations?
3. How do logarithmic transformations help simplify exponential equations?
4. What is the significance of squaring equations in eliminating square roots?
5. What role do natural logarithms play in solving equations involving the exponential function?

### Tip

When solving systems of equations, try to isolate one variable and substitute it into the other equation to reduce the system to a single-variable equation.

Learn how to solve a system of equations involving exponential and logarithmic transformations. Detailed step-by-step solution for finding values of x and y satisfying two given equations. Suitable for advanced mathematics students.

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