Maximise Z = 3 x + 4 y Subject to the constraints:

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Solution

The given constraints are,

x+y≤4 x≥0 y≥0

The objective function which needs to maximize is given as,

Z=3x+4y

The line x+y≤4 gives the intersection point as,

x 0 4
y 4 0

Also, when x=0,y=0 for the line x+y≤4, then,

0+0≤4 0≤4

This is true, so the graph have the shaded region towards the origin.

Plot the points of all the constraint lines.

It can be observed that the corner points are O( 0,0 ),A( 4,0 ),B( 0,4 ).

Substitute these points in the given objective function to find the maximum value of Z.

Corner points Z=3x+4y
O( 0,0 ) 0
A( 4,0 ) 12
B( 0,4 ) 16 (Maximum)

Therefore, the maximum value of Z is 16 at the point ( 0,4 ).

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