1. The problem is to convert the polar coordinate $\left(7, \frac{3\pi}{4}\right)$ to Cartesian coordinates $(x,y)$. 2. The formulas to convert from polar to Cartesian coordinates are: $$x = r \cos \theta$$ $$y = r \sin \theta$$ where $r$ is the radius and $\theta$ is the angle. 3. Given $r = 7$ and $\theta = \frac{3\pi}{4}$, substitute these values into the formulas: $$x = 7 \cos \frac{3\pi}{4}$$ $$y = 7 \sin \frac{3\pi}{4}$$ 4. Evaluate the trigonometric functions: $$\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$$ $$\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}$$ 5. Multiply by $7$: $$x = 7 \times -\frac{\sqrt{2}}{2} = -\frac{7\sqrt{2}}{2}$$ $$y = 7 \times \frac{\sqrt{2}}{2} = \frac{7\sqrt{2}}{2}$$ 6. Therefore, the Cartesian coordinates are: $$\boxed{\left(-\frac{7\sqrt{2}}{2}, \frac{7\sqrt{2}}{2}\right)}$$