Find the points $(x_i, y_i)$ at which each pair of equations intersect. There are three such pairs, so three points of intersection, which will be the vertices of the triangle. From the vertices $(x_i, y_i)$, you can determine the two vectors you need, which you can use as the columns of the matrix for which the absolute value of the determinant, multiplied by 1/2, will give you area.
Given $\;y = x$, $y = -3x+8$, and $3y + 5x = 0$
Vertex 1: At what point does $y = x$ and $y = -3x + 8$ intersect? When $x = -3x + 8$. Solving for $x$, gives us $x = 2$, which in this case, will also equal $y$.
So vertex 1 is $(2, 2)$. Proceed in a similar manner to determine:
Vertex 2: where $y = x$ and $3y + 5x = 0$.
Vertex 3: where $y - -3x + 8$ and $3y + 5x = 0$