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The question asks to determine the position of the line \( y = 3x - 1 \) relative to the circle given by the equation:

\[
x^2 + y^2 + 2x + 2y - 4 = 0.
\]

### Step 1: Rewrite the circle equation in standard form
Let's first rewrite the equation of the circle in a more standard form. We complete the square for both \( x \) and \( y \) terms.

1. **Group the \( x \) and \( y \) terms:**

   \[
   x^2 + 2x + y^2 + 2y - 4 = 0.
   \]

2. **Complete the square for \( x \):**
   \[
   x^2 + 2x = (x + 1)^2 - 1.
   \]

3. **Complete the square for \( y \):**
   \[
   y^2 + 2y = (y + 1)^2 - 1.
   \]

4. **Substitute these into the original equation:**

   \[
   (x + 1)^2 - 1 + (y + 1)^2 - 1 - 4 = 0,
   \]
   \[
   (x + 1)^2 + (y + 1)^2 - 6 = 0,
   \]
   \[
   (x + 1)^2 + (y + 1)^2 = 6.
   \]

This is the equation of a circle with center at \( (-1, -1) \) and radius \( \sqrt{6} \).

### Step 2: Find the distance between the center of the circle and the line
The formula for the distance \( d \) from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \) is given by:

\[
d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}.
\]

For the line \( y = 3x - 1 \), we can rewrite it in the form:

\[
3x - y - 1 = 0.
\]

Here, \( A = 3 \), \( B = -1 \), and \( C = -1 \).

Substitute the center of the circle \( (-1, -1) \) into the formula:

\[
d = \frac{|3(-1) - 1(-1) - 1|}{\sqrt{3^2 + (-1)^2}} = \frac{|-3 + 1 - 1|}{\sqrt{9 + 1}} = \frac{|-3|}{\sqrt{10}} = \frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{10}.
\]

### Step 3: Compare the distance with the radius
The radius of the circle is \( \sqrt{6} \).

- If \( d > \sqrt{6} \), the line is outside the circle.
- If \( d = \sqrt{6} \), the line is tangent to the circle.
- If \( d < \sqrt{6} \), the line intersects the circle.

Calculating \( \sqrt{6} \approx 2.45 \).

Since \( \frac{3\sqrt{10}}{10} \approx 0.95 < \sqrt{6} \), the line intersects the circle.

### Conclusion
The line \( y = 3x - 1 \) intersects the circle \( x^2 + y^2 + 2x + 2y - 4 = 0 \).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the points of intersection between the line and the circle?
2. What is the general method for finding the position of a line relative to a circle?
3. Can the same approach be used for determining the position of a line relative to an ellipse?
4. How do you find the tangent line to a circle from a given point outside the circle?
5. How would the solution change if the line were vertical or horizontal?

**Tip:** Always check your line and circle equations for possible simplifications before performing calculations.

Tentukan posisi garis y = 3x - 1 terhadap lingkaran x^2 + y^2 + 2x + 2y - 4 = 0!

This problem involves determining the relationship between a line and a circle by applying concepts such as the equation of a circle and the distance formula. Learn how to solve it step by step, including rewriting the circle in standard form and calculating the distance from the center of the circle to the line.

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