The equations are:
x=0 (y-axis)
y=0 (x-axis)
2x+4y=20 or x+2y=10
3x+2y=24
The easiest way to find the minimum value of Z given the constraints is to draw a graph and find out what geometrical figure represents the optimization problem. In this case, the figure is a triangle with vertices at (0,5), (0,12) and (7,1.5) (see below).
Solving the system of equations: 2x+4y=20, 3x+2y=24.
The first equation reduces to x+2y=10, so 2y=10-x, and 3x+10-x=24, 2x=14, x=7, y=1.5. This gives us the vertex (7,1.5). The triangle represents the feasible region defined by the constraints. If we plug the coordinates of the vertices into Z, we get respectively 4x+6y=30, 72, 28+9=37. Clearly 30 is the smallest value, so x=0 and y=5.