Solve this equation. If the equation has no valid solutions, click on the appropriate button. \[ \sqrt{7 x-2}=\sqrt{5 x+1} \]

To solve the equation \(\sqrt{7x - 2} = \sqrt{5x + 1}\), we start by squaring both sides to eliminate the square roots: \[ 7x - 2 = 5x + 1 \] Next, we simplify the equation by moving all terms involving \(x\) to one side and constant terms to the other side: \[ 7x - 5x = 1 + 2 \] \[ 2x = 3 \] Now, we solve for \(x\): \[ x = \frac{3}{2} \] To confirm that this solution is valid, we substitute \(x = \frac{3}{2}\) back into the original equation: \[ \sqrt{7\left(\frac{3}{2}\right) - 2} = \sqrt{5\left(\frac{3}{2}\right) + 1} \] \[ \sqrt{\frac{21}{2} - 2} = \sqrt{\frac{15}{2} + 1} \] \[ \sqrt{\frac{21}{2} - \frac{4}{2}} = \sqrt{\frac{15}{2} + \frac{2}{2}} \] \[ \sqrt{\frac{17}{2}} = \sqrt{\frac{17}{2}} \] Since both sides are equal, the solution is valid. Thus, the solution to the equation is: \[ x = \frac{3}{2} \]