Minimize `Z=4x+5y` subject to `2x+y ge 7, 2x+3y le 15, x le 3, x ge0, y ge0` solve using graphical method.

Explanation

Corner point Method :
Step 1: Convert the inequations into the equations and draw the corresponding lines.
`2x+y=7implies(x)/(7//2)+y/7=1`
`2x+3y=15implies(x)/(15//2)+y/5=1`
`x=3`
and `x ge0, y ge0,` the solution lies in the first quadrant.
`therefore` ABCD is the feasible region.
Step 2: We find the coordinates of the vertices A, B, C
`A=(3,1)`
`B(3,3)`
`C((3)/(2),4)`

Step 3: We find the values of the objective function, `Z=4x+5y` at the corner points (vertices) A, B and C respectively.
`Z(A)=4(3)+5(1)=12+5=17`
`Z(B)=4(3)+5(3) =12+15=27`
`Z(C)=4((3)/(2))+5(4)=6+20=26.`
`therefore` Z has minimum value 17 at (3,1).
`therefore `Optimum solution is `x=3, y=1` with minimum value.