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To find the equation of a line \( l \) passing through the point \( P(4,7) \) with an inclination angle \( v = 30^\circ \), we can use the following approach:

1. **Calculate the slope \( m \) using the inclination angle**:
   The slope of a line with an inclination angle \( v \) is given by:
   \[
   m = \tan(v)
   \]
   Given \( v = 30^\circ \), we find:
   \[
   m = \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577
   \]

2. **Form the equation of the line**:
   The equation of a line in point-slope form is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \( P(4,7) \) as \( (x_1, y_1) \) and \( m = \frac{1}{\sqrt{3}} \), we get:
   \[
   y - 7 = \frac{1}{\sqrt{3}} (x - 4)
   \]

3. **Simplify the equation (optional)**:
   Expanding and simplifying the equation:
   \[
   y = \frac{1}{\sqrt{3}} x - \frac{4}{\sqrt{3}} + 7
   \]
   Or, in a more approximate form:
   \[
   y \approx 0.577 x + 4.31
   \]

Thus, the equation of the line \( l \) is:
\[
y = \frac{1}{\sqrt{3}} x + 4.31
\]

Would you like a deeper explanation or more details on this solution? 

Here are some related questions you might explore:

1. How would the equation change if the inclination angle was different?
2. What is the effect of using a different point on the line to form the equation?
3. How do we interpret the y-intercept of this line?
4. How can we confirm if another point lies on this line?
5. What would the line equation be in a different form, such as the general form \( Ax + By + C = 0 \)?

**Tip:** The slope of a line directly depends on the angle it makes with the positive x-axis; this is particularly useful for visualizing lines in geometry.

En linje l går gennem punktet P(4,7) og har hældningsvinklen v = 30°. a) Bestem en ligning for l.

Find the equation of a line passing through point P(4,7) with an inclination angle of 30°. This problem explores concepts in analytic geometry and includes step-by-step calculations for determining the slope using the angle and applying the point-slope form. Suitable for high school students in Grades 10-12.

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