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To find the point on the line \( y = 5x + 4 \) that is closest to the origin, we can follow these steps:

1. **Set Up the Distance Formula:** The distance from a point \( (x, y) \) on the line to the origin \( (0, 0) \) is given by:
   \[
   d = \sqrt{x^2 + y^2}
   \]
   Since \( y = 5x + 4 \), we can substitute \( y \) in terms of \( x \):
   \[
   d = \sqrt{x^2 + (5x + 4)^2}
   \]

2. **Minimize the Distance:** To find the point closest to the origin, we need to minimize \( d^2 \) (which is equivalent to minimizing \( d \) since the square root is a monotonically increasing function):
   \[
   d^2 = x^2 + (5x + 4)^2
   \]
   Expanding this:
   \[
   d^2 = x^2 + (25x^2 + 40x + 16)
   \]
   \[
   d^2 = 26x^2 + 40x + 16
   \]

3. **Take the Derivative and Set It to Zero:** Now, differentiate \( d^2 \) with respect to \( x \) and set the derivative equal to zero to find the minimum.
   \[
   \frac{d(d^2)}{dx} = 52x + 40 = 0
   \]
   Solving for \( x \):
   \[
   x = -\frac{40}{52} = -\frac{10}{13}
   \]

4. **Find the Corresponding \( y \)-Value:** Substitute \( x = -\frac{10}{13} \) back into the line equation \( y = 5x + 4 \):
   \[
   y = 5\left(-\frac{10}{13}\right) + 4 = -\frac{50}{13} + \frac{52}{13} = \frac{2}{13}
   \]

5. **Solution:** The point on the line \( y = 5x + 4 \) that is closest to the origin is:
   \[
   \left( -\frac{10}{13}, \frac{2}{13} \right)
   \]

Would you like more details on any step?

### Related Questions
1. How do you derive the distance formula for a point and the origin?
2. Why do we use the derivative to find minimum distances?
3. What would change if the line equation had a different slope?
4. How can we generalize this method for any line equation?
5. Why does minimizing \( d^2 \) instead of \( d \) work?

### Tip
In optimization problems involving distance, minimizing \( d^2 \) is often simpler than working directly with \( d \), as it avoids the square root.

Find the point on the line y = 5x + 4 that is closest to the origin.

This problem demonstrates how to find the point on the line y = 5x + 4 that is closest to the origin by using the distance formula and optimization techniques. It includes a step-by-step approach to minimize the distance to the origin using calculus. Suitable for high school students in Grades 10-12.

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