Question

Find the graphical solution for the following system of linear equations :

3x + 4y ≥ 12 , 4x + 7y ≤ 28 , y ≥ 1 , x ≥ 0 , y ≥ 0 ,

Hence find the co-ordinates of comer points of the common region. 

Answer 1

In equation	Corresponding equation	Points on
		X-axis	Y-axis
1. 3x + 4y ≥  12	3x + 4y = 12	A(4 , 0)	B(0 , 3)
2.4x + 7y ≤ 28	4x + 7y = 28	C (7 , 0)	D(0 , 4)
3. y ≥ 1	y = 1	-	E(0 ,1)
4. x ≥ 0 , y ≥ 0	x = 0 , y=0	-	-

From the graph the shaded region FGDB is the feasible region.

The objective function will be optimum at the vertices.

From the diagram F is the point of intersection of the lines 3x+4y= 12 and y = 1.

By substituting y= 1, we will get 

3x + 4(1) = 12 ⇒ x = `8/3`

∴ F = `(8/3 , 1)`

Also G is the point of intersection of the

lines 4x+7y=28 and y= 1 

By substituting y= 1, we wm get 

4x + 7(1) = 28 ⇒ 4x = 28 - 7 

∴ X = `21/4`

∴ G = `(21/4 , 1)`

∴  Co-ordinates of the feasible region are 

F = `(8/3 , 1)` , G = `(21/4 , 1)` , D (0 , 4) and B (0 , 3)