What point on the line $y=5 x+4$ is closest to the origin?

The point on the line $y=5 x+4$ closest to the origin is
(Type an ordered pair using integers or fractions. Simplify your answer.)

Step 1 ：We are given the line equation \(y=5x+4\) and we need to find the point on this line that is closest to the origin.

Step 2 ：The distance between a point \((x, y)\) and the origin is given by the formula \(\sqrt{x^2 + y^2}\). In this case, \(y = 5x + 4\), so we want to minimize the function \(f(x) = \sqrt{x^2 + (5x + 4)^2}\).

Step 3 ：We can find the minimum of this function by taking its derivative and setting it equal to zero. The x-coordinate of the point where this occurs will give us the x-coordinate of the point on the line that is closest to the origin.

Step 4 ：The derivative of \(f(x)\) is \(\frac{26x + 20}{\sqrt{x^2 + (5x + 4)^2}}\). Setting this equal to zero, we find that \(x = -\frac{10}{13}\).

Step 5 ：We can then substitute this x-coordinate back into the equation of the line to find the corresponding y-coordinate. Substituting \(x = -\frac{10}{13}\) into \(y = 5x + 4\), we find that \(y = \frac{2}{13}\).

Step 6 ：Final Answer: The point on the line \(y=5 x+4\) that is closest to the origin is \(\boxed{\left(-\frac{10}{13}, \frac{2}{13}\right)}\).