9 Find the point on the curve $2 x+3 y=7$ that is closest to the origin.

Step 1 ：We are given the line equation \(2x + 3y = 7\) and we need to find the point on this line that is closest to the origin.

Step 2 ：We can express \(y\) in terms of \(x\) from the equation of the line as \(y = \frac{7}{3} - \frac{2x}{3}\).

Step 3 ：We can then substitute this into the distance formula, \(\sqrt{x^2 + y^2}\), to get the distance from the origin to any point on the line as a function of \(x\): \(\sqrt{x^2 + \left(\frac{7}{3} - \frac{2x}{3}\right)^2}\).

Step 4 ：We can then take the derivative of this distance with respect to \(x\) to find the \(x\)-value that minimizes the distance. The derivative is \(\frac{13x}{9} - \frac{14}{9}\) divided by the square root of \(x^2 + \left(\frac{7}{3} - \frac{2x}{3}\right)^2\).

Step 5 ：Setting this derivative equal to zero and solving for \(x\), we find that the \(x\)-value that minimizes the distance is \(\frac{14}{13}\).

Step 6 ：We can then substitute this \(x\)-value back into the equation of the line to find the corresponding \(y\)-value, which is \(\frac{21}{13}\).

Step 7 ：\(\boxed{\left(\frac{14}{13}, \frac{21}{13}\right)}\) is the point on the curve \(2x + 3y = 7\) that is closest to the origin.