Find the point on the line $y=2 x+3$ that is closest to the origin. \[ (x, y)=(\square) \]

Step 1 ：We are given the line equation \(y = 2x + 3\) 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 \(\sqrt{x^2 + y^2}\). However, since the square root function is monotonically increasing, we can instead minimize the square of the distance, \(x^2 + y^2\).

Step 3 ：We substitute \(y = 2x + 3\) into our distance formula to get \(x^2 + (2x + 3)^2\).

Step 4 ：We then take the derivative of this function with respect to \(x\) and set it equal to zero to find the minimum. The derivative is \(10x + 12\).

Step 5 ：The critical point is at \(x = -\frac{6}{5}\). We substitute this value back into the equation \(y = 2x + 3\) to find the corresponding \(y\) value, which is \(\frac{3}{5}\).

Step 6 ：\(\boxed{\text{Final Answer: The point on the line } y=2x+3 \text{ that is closest to the origin is } \left(-\frac{6}{5}, \frac{3}{5}\right)}\)