1. **Stating the problem:** We are given two determinants and asked to find the values of $x$ and $y$ defined as $$x = \frac{1}{5} \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix}, \quad y = \frac{1}{5} \begin{vmatrix} 7 & a \\ b & c \end{vmatrix}$$ 2. **Formula for determinant of a 2x2 matrix:** For a matrix $$\begin{pmatrix} p & q \\ r & s \end{pmatrix},$$ the determinant is $$\det = ps - rq.$$ 3. **Calculate $x$:** $$\det_x = (1)(4) - (2)(3) = 4 - 6 = -2.$$ Therefore, $$x = \frac{1}{5} \times (-2) = -\frac{2}{5}.$$ 4. **Calculate $y$:** $$\det_y = (7)(c) - (a)(b) = 7c - ab.$$ Therefore, $$y = \frac{1}{5} (7c - ab) = \frac{7c - ab}{5}.$$ 5. **Summary:** - $x = -\frac{2}{5}$ - $y = \frac{7c - ab}{5}$ These results express $x$ as a number and $y$ in terms of variables $a,b,c$.