Hello
1. I've modified your sketch a little bit (see attachment): I refer to the labels of my sketch.
2. Use Pythagorian theorem and Cosine rule: You'll get 4 equations:
- $\displaystyle d^2=7^2+y^2$
- $\displaystyle d^2=x^2+4^2-2\cdot 4 \cdot x \cdot \cos(60^\circ)$
- $\displaystyle s^2=4^2+y^2-2 \cdot 4 \cdot y \cdot \cos(150^\circ)$
- $\displaystyle s^2=x^2+7^2-2 \cdot 7 \cdot x \cdot \cos(60^\circ)$
3. With $\displaystyle \cos(60^\circ)=\frac12$ and $\displaystyle \cos(150^\circ)=-\frac12 \cdot \sqrt{3}$ you'll get rid of the trigonometric functions.
4. The calculations are really ugly und you'll get many opportunities to make mistakes. For conformation only:
$\displaystyle (x,y,d,s)=(10,3\sqrt{3},2\sqrt{19},\sqrt{79})$