# Class 12 RD Sharma Solutions – Chapter 30 Linear Programming – Exercise 30.2 | Set 2

**Question 11. ****Minimize Z = 5x + 3y**

**Subject to**

**2x + y ****≥ 10**

**x + 3y ≥ 15**

**x ≤ 10**

**y ≤ 8**

**x, y ≥ 0**

**Solution:**

Convert the given in equations into equations, we will get the following equations:

2x + y = 10, x + 3y = 15, x = 10, y = 8

Area shown by 2x + y ≥ 10:

The line 2x + y = 10 meets the coordinate axes at A(5, 0) and B(0, 10) respectively.

After connecting these points we will get the line 2x + y = 10.

Thus,

(0,0) does not assure the in equation 2x + y ≥ 10.

Thus,

The area in xy plane which does not have the origin represents the solution set of the in equation 2x + y ≥ 10.

The area represented by x + 3y ≥ 15:

The line x + 3y = 15 connects the coordinate axes at C(15, 0) and D(0, 5) respectively.

After connecting these points we will get the line x + 3y = 15.

Thus,

(0,0) assure the in equation x + 3y ≥ 15.

The area in xy plane which does not have the origin represents the solution set of the in equation x + 3y ≥ 15.

The line x = 10 is the line that passes through the point (10, 0) and is parallel to Y axis.x ≤ 10 is the area to the left of the line

x = 10.

The line y = 8 is the line that passes through the point (0, 8) and is parallel to X axis.y ≤ 8 is the area below the line y = 8.

The area shows by x ≥ 0 and y ≥ 0:

Thus,

All point in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shows by the in equations x ≥ 0 and y ≥ 0.

The suitable area determined by the system of constraints, 2x + y ≥ 10, x + 3y ≥ 15, x ≤ 10, y ≤ 8, x ≥ 0 and y ≥ 0 are as

follows.

The corner points of the suitable region are

E(3, 4),

H,

F(10, 8) and

G(1, 8).

The values of Z at these corner points are as follows.

Corner point Z = 5

x+ 3y

E(3, 4)5 × 3 + 3 × 4 = 27

H(10,53)

5 × 10 + 3× = 55

F(10, 8)5 × 10 + 3 × 8 = 74

G(1, 8)5 × 1 + 3 × 8 = 29

Hence,

The minimum value of Z is 27 at the point F(3, 4).

Therefore,

x = 3 and y =4 is the best solution of the given LPP.

Hence, the best value of Z is 27.

**Question 12. ****Minimize Z = 30x + 20y**

**Subject to**

**x + y ****≤ 8**

**x + 4y ≥ 12**

**5x + 8y = 20**

**x, y ≥ 0**

**Solution:**

Convert the given in equations into equations, we will get the following equations:

x + y = 8, x + 4y = 12, x = 0 and y = 0

5x + 8y = 20 is already an equation.

The area shows by x + y ≤ 8:

The line x + y = 8 connects the coordinate axes at A(8, 0) and B(0, 8) respectively.

After connecting these points we will get the line x + y = 8.

Thus,

(0,0) assures the in equation x + y ≤ 8.

Thus,

The area in xy plane which have the origin represents the solution set of the in equation x + y ≤ 8.

The area shows by x + 4y ≥ 12:

The line x + 4y = 12 connects the coordinate axes at C(12, 0) and D(0, 3) respectively.

After connecting these points we will get the line x + 4y = 12.

Thus,

(0,0) assure the in equation x + 4y ≥ 12.

Thus,

The area in xy plane which does not have the origin shows the solution set of the in equation x + 4y ≥ 12.

The line 5x + 8y = 20 is the line that passes through E(4, 0) and F .

The area shows by x ≥ 0 and y ≥ 0:

Thus,

All point in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shows by the in equations x ≥ 0 and y ≥ 0.

The suitable area determined by the system of constraints, x + y ≤ 8, x + 4y ≥ 12, 5x + 8y = 20, x ≥ 0 and y ≥ 0 are as follows.

The corner points of the suitable area are B(0,8), D(0,3), G .

The values of Z at these corner points are as follows.

Corner point Z = 30

x+ 20yB(0,8)

160

D(0,3)

60

G

266.66

Hence,

The minimum value of Z is 60 at the point D(0,3).

Therefore,

x = 0 and y =3 is the best solution of the given LPP.

Therefore,

The best value of Z is 60.

**Question 13. Maximize Z = 4x + 3y**

**Subject to**

**3x + 4y ≤ 24**

**8x + 6y ≤ 48**

** x ≤ 6000**

** x ≤ y**

** x, y ≥ 0**

**Solution:**

Here, we need to maximize Z = 4x + 3y

Convert the given in equations into equations, we will get the following equations:

3x + 4y = 24, 8x + 6y = 48, x = 5 , y = 6, x = 0 and y = 0.

The line 3x + 4y = 24 connects the coordinate axis at A(8, 0) and B(0,6).

Connects these points to get the line 3x + 4y = 24.

Thus,

(0, 0) assure the in equation 3x + 4y ≤ 24.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

The line 8x + 6y = 48 connects the coordinate axis at C(6, 0) and D(0,8).

Connect these points to get the line 8x + 6y = 48.

Thus,

(0, 0) assure the in equation 8x + 6y ≤ 48.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

x = 5 is the line passing through x = 5 parallel to the Y axis.

y = 6 is the line passing through y = 6 parallel to the X axis.

The area shown by x ≥ 0 and y ≥ 0:

Hence,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations.

These lines are drawn using a suitable scale.

The corner points of the feasible region are O(0, 0), G(5, 0), F , E and B(0, 6).

The values of Z at these corner points are as follows.

Corner point

Z = 4

x+ 3yO(0, 0)

4× 0 + 3 × 0 = 0

G(5, 0) G5, 0

4 × 5 + 3 × 0 = 20

F

4 × 5 + 3 × = 24

E

4 × + 3 × = = 24

B(0, 6)B0, 6

4 × 0 + 3 × 6 = 18

Here we can see that the maximum value of the objective function Z is 24 which is at F and E .

Therefore,

The best value of Z is 24.

**Question 14. ****Minimize Z = x – 5y + 20**

**Subject to**

** x – y ****≥ 0**

**-x + 2y ≥ 2**

** x ≥ 3**

** y ≤ 4**

** x, y ≥ 0**

**Solution:**

Convert the given in equations into equations, we will get the following equations:

x − y = 0, − x + 2y = 2, x = 3, y = 4, x = 0 and y = 0.

The area shown by x − y ≥ 0 or x ≥ y:

The line x − y = 0 or x = y passes through the origin.The area to the right of the line x = y will assure the given in equation.

Now we will check by taking an example like if we take a point (4, 3) to the right of the line x = y .

Here, x ≥ y.

Thus,

It assure the given in equation.

Take a point (4, 5) to the left of the line x = y. Here, x ≤ y. That means it does not assure the given in equation.

The area shown by − x + 2y ≥ 2:The line − x + 2y = 2 connects the coordinate axes at A(−2, 0) and B(0, 1) respectively.

After connecting these points we will get the line − x + 2y = 2.

Thus,

(0,0) does not assure the in equation − x + 2y ≥ 2.

Thus,

The area in xy plane which does not have the origin represents the solution set of the in equation − x + 2y ≥ 2 .

The line x = 3 is the line that passes through the point (3, 0) and is parallel to Y axis. x ≥ 3 is the area to the right of the

line x = 3.

The line y = 4 is the line that passes through the point (0, 4) and is parallel to X axis. y ≤ 4 is the area below the line y = 4.

The area shown by x ≥ 0 and y ≥ 0:Hence,

All point in the first quadrant assure these in equations.

Thus, the first quadrant is the area shown by the in equations x ≥ 0 and y ≥ 0.

The suitable area determined by the system of constraints x − y ≥ 0,− x + 2y ≥ 2, x ≥ 3, y ≤ 4, x ≥ 0 and y ≥ 0 are as follows.

The corner points of the suitable area are C , D(3, 3), E(4, 4) and F(6, 4).

The values of Z at these corner points are as follows.

Corner point

Z =

x− 5y+ 20C

3 − 5 × + 20 =

D(3, 3)D3, 3

3 − 5 × 3 + 20 = 8

E(4, 4)4 − 5 × 4 + 20 = 4

F(6, 4)6 − 5 × 4 + 20 = 6

Hence,

The minimum value of Z is 4 at the point E(4, 4).

Therefore,

x = 4 and y = 4 is the best solution of the given LPP.

Hence,

The best value of Z is 4.

**Question 15. Maximize Z = 3x + 5y**

**Subject to**

**x + 2y ≤ 20**

** x + y ≤ 15**

** y ≤ 15**

** x, y ≥ 0**

**Solution:**

Here we need to maximize Z = 3x + 5y

Convert the given in equations into equations, we will get the following equations:

x + 2y = 20, x + y = 15, y = 5, x = 0 and y = 0.

The line x + 2y = 20 connects the coordinate axis at A(20, 0) and B(0,10).

Connect these points to get the line x + 2y = 20.

Thus,

(0, 0) assume the in equation x + 2y ≤ 20.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

The line x + y = 15 connect the coordinate axis at C(15, 0) and D(0,15).

Connect these points to get the line x + y = 15.

Thus,

(0, 0) assume the in equation x + y ≤ 15.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

y = 5 is the line passing through (0, 5) and parallel to the X axis. The area below the line y = 5 will assure the given in equation.

The area shown by x ≥ 0 and y ≥ 0:

Hence,

All the point in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations.

These lines are drawn using a suitable scale.

The corner points of the suitable area are O(0, 0), C(15, 0), E(10, 5) and F(0, 5)

The values of Z at these corner points are as follows.

Corner point

Z = 3

x+ 5y

O(0, 0)3 × 0 + 5 × 0 = 0

C(15, 0)

3 × 15 + 5 × 0 = 45

E(10, 5)

3 × 10 + 5 × 5 = 55

F(0, 5)

3 × 0 + 5 × 5 = 25

Here we can see that the maximum value of the objective function Z is 55 which is at E(10, 5).

Therefore,

The best value of Z is 55.

**Question 16. ****Minimize Z = 3x**_{1} + 5x_{2}

_{1}+ 5x

_{2}

**Subject to**

**x _{1} + x_{2} **

**≥ 3**

**x _{1} + x_{2} ≥ 2**

**x _{1}, x_{2} ≥ 0**

**Solution:**

Convert the given in equations into equations, we will get the following equations:

x

_{1}+ 3x_{2}= 3, x_{1}+ x_{2}= 2, x_{1}= 0 and x_{2}= 0The area shown by x

_{1}+ 3x_{2}≥ 3 :The line x

_{1}+ 3x_{2}= 3 connects the coordinate axes at A(3, 0) and B(0, 1) respectively.After connecting these points we will get the line x

_{1}+ 3x_{2}= 3.Thus,

(0,0) does not assure the in equation x

_{1}+ 3x_{2}≥ 3.Thus,

The area in the plane which does not have the origin represents the solution set of the in equation x

_{1}+ 3x_{2}≥ 3.The area shown by x

_{1}+ x_{2}≥ 2:The line x

_{1}+ x_{2}= 2 connects the coordinate axes at C(2, 0) and D(0, 2) respectively.After connecting these points we will get the line x

_{1}+ x_{2}= 2.Thus,

(0,0) does not assure the in equation x

_{1}+ x_{2}≥ 2.Thus,

The area having the origin represents the solution set of the in equation x

_{1}+ x_{2}≥ 2.The area shown by x

_{1}≥ 0 and x_{2}≥ 0:Hence,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations x

_{1}≥ 0 and x_{2}≥ 0.The suitable area determined by the system of constraints, x

_{1}+ 3x_{2}≥ 3 , x_{1}+ x_{2}≥ 2, x_{1}≥ 0, and x_{2}≥ 0, are as follows.The corner points of the suitable area are O(0, 0), B(0, 1), E and C(2, 0).

The values of Z at these corner points are as follows.

Corner point

Z = 3

x1 + 5x2

O(0, 0)3 × 0 + 5 × 0 = 0

B(0, 1)3 × 0 + 5 × 1 = 5

E

3 × + 5 × = 7

C(2, 0)3 × 2 + 5 × 0 = 6

Hence,

The minimum value of Z is 0 at the point O(0, 0).

Hence,

x

_{1}= 0 and x_{2}= 0 is the best solution of the given LPP.Therefore,

The best value of Z is 0.

**Question 17. Maximize Z = 2x + 3y**

**Subject to**

** x + y ≥ 1**

** 10x + y ≥ 5**

** x + 10y ≥ 1**

** x, y ≥ 0**

**Solution:**

Convert the given in equations into equations, we will get the following equations:

x + y = 1, 10x +y = 5, x + 10y = 1, x = 0 and y = 0

The area shown by x + y ≥ 1:

The line x + y = 1 connects the coordinate axes at A(1, 0) and B(0,1) respectively.

After connecting these points we will get the line x + y = 1.

Thus,

(0,0) does not assure the in equation x + y ≥ 1.

Thus,

The area in xy plane which does not have the origin represents the solution set of the in equation x + y ≥ 1.

The area shown by 10x +y ≥ 5:

The line 10x +y = 5 connect the coordinate axes at C and D(0, 5) respectively.

After connecting these points we will get the line 10x +y = 5.

Thus,

(0,0) does not assure the in equation 10x +y ≥ 5.

Thus, the area which does not have the origin represents the solution set of the in equation 10x +y ≥ 5.

The area shown by x + 10y ≥ 1:

The line x + 10y = 1 connect the coordinate axes at A(1, 0) and F respectively.

After connecting these points we will have the line x + 10y = 1.

Thus,

(0,0) does not assure the in equation x + 10y ≥ 1.

Thus, the area which does not have the origin represents the solution set of the in equation x + 10y ≥ 1.

The area shown by x ≥ 0 and y ≥ 0:

Here,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations x ≥ 0, and y ≥ 0.

The suitable area determined by the system of constraints x + y ≥ 1, 10x +y ≥ 5, x + 10y ≥ 1, x ≥ 0, and y ≥ 0, are as follows.

The suitable area is unbounded.

Hence,

The maximum value is infinity i.e. the solution is unbounded.

**Question 18. ****Maximize Z = -x**_{1} + 2x_{2}

_{1}+ 2x

_{2}

**Subject to**

** -x _{1} + 3x_{2} ≤ 10**

** x _{1} + x_{2} ≤ 6**

** x _{1} – x_{2} ≤ 2**

** x _{1}, x_{2} ≥ 0**

**Solution:**

Convert the given in equations into equations, we will get the following equations:

−x

_{1}+ 3x_{2}= 10, x_{1}+ x_{2}= 6, x_{1}+ x_{2}= 2, x_{1}= 0 and x_{2}= 0The area shown by −x

_{1}+ 3x_{2}≤ 10:The line −x

_{1}+ 3x_{2}= 10 coincide the coordinate axes at A(−10, 0) and B respectively.After connecting these points we will get the line −x

_{1}+ 3x_{2}= 10.Thus,

(0,0) satisfies the in equation −x

_{1}+ 3x_{2}≤ 10 .Thus,

The area region in the plane which have the origin shows the solution set of the in equation

−x

_{1}+ 3x_{2}≤ 10.The area shown by x

_{1}+ x_{2}≤ 6:The line x

_{1}+ x_{2}= 6 connects the coordinate axes at C(6, 0) and D(0, 6) respectively.After connecting these points we will get the line x

_{1}+ x_{2}= 6.Thus,

(0,0) assure the in equation x

_{1}+ x_{2}≤ 6.Thus,

The area having the origin represents the solution set of the in equation x

_{1}+ x_{2}≤ 6.The area shown by x

_{1}− x_{2}≤ 2:The line x

_{1}− x_{2}= 2 coincide the coordinate axes at E(2, 0) and F(0, −2) respectively.After connecting these points we will get the line x

_{1}− x_{2}= 2.Thus,

(0,0) assure the in equation x

_{1}− x_{2}≤ 2.Thus,

The area having the origin represents the solution set of the in equation x

_{1}− x_{2}≤ 2.The area shown by x

_{1}≥ 0 and x_{2}≥ 0:Hence,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the region shown by the in equations x

_{1}≥ 0 and x_{2}≥ 0.The suitable area determined by the system of constraints, −x

_{1}+ 3x_{2}≤ 10, x_{1}+ x_{2}≤ 6, x_{1}− x_{2}≤ 2, x_{1}≥ 0, and x_{2}≥ 0, are as follows.The corner points of the assure area are O(0, 0), E(2, 0), H(4, 2), G(2, 4) and B .

The values of Z at these corner points are as follows.

Corner point

Z =

−x+ 2_{1}x_{2}O(0, 0)

−1 × 0 + 2 × 0 = 0

E(2, 0)

−1 × 2 + 2 × 0 =−2

H(4, 2)

−1 × 4 + 2 × 2 = 0G(2, 4)

−1 × 2 + 2 × 4 = 6B

−1 × 0 + 2 × =Here we can see that the maximum value of the objective function Z is which is at B .

**Question 19. ****Maximize Z = -x + y**

**Subject to**

**-2x + y ≤ 1**

** x ≤ 2**

** x + y ≤ 3**

** x, x ≥ 0**

**Solution:**

Here we need to maximize Z = x + y

Convert the given in equations into equations, we will get the following equations:

−2x + y = 1, x = 2, x + y = 3, x = 0 and y = 0.

The line −2x + y = 1 coincide the coordinate axis at A and B(0, 1).

Now connect these points to get the line −2x + y = 1 .

Thus,

(0, 0) assure the in equation −2x + y ≤ 1.

Thus,

The area in xy-plane that have the origin represents the solution set of the given equation.

x = 2 is the line passing through (2, 0) and parallel to the Y axis.

The area below the line x = 2 will assure the given in equation.

The line x + y = 3 coincide the coordinate axis at C(3, 0) and D(0, 3).

Connect these points to get the line x + y = 3.

Thus,

(0, 0) assure the in equation x + y ≤ 3.

Thus,

The area in xy-plane that have the origin shows the solution set of the given equation.

The area shown by x ≥ 0 and y ≥ 0:

Hence,

All the points in the first quadrant assures these in equations.

Thus,

The first quadrant is the area shown by the in equations.

These lines are drawn using a satisfactory scale.

The corner points of the suitable area are O(0, 0),G(2, 0), E(2, 1) and F

The values of Z at these corner points are as follows.

Corner point

Z =

x+y

O(0, 0)0 + 0 = 0

C(2, 0)

2 + 0 = 2

E(2, 1)

2 +1 = 3

F

Here we can see that the maximum value of the objective function Z is 3 which is at E(2, 1) and F .

Therefore,

The best value of Z is 3.

**Question 20. **** ****Maximize Z = -3x**_{1} + 4x_{2}

_{1}+ 4x

_{2}

**Subject to**

** x _{1} – x_{2} ≤ -1**

** x _{1} + x_{2} ≤ 0**

** x _{1}, x_{2} ≥ 0**

**Solution:**

Convert the given in equations into equations, we will get the following equations:

x

_{1}− x_{2}= −1, −x_{1}+ x_{2}= 0, x_{1}= 0 and x_{2}= 0The area shown by x

_{1}− x_{2}≤ −1:The line x

_{1}− x_{2}= −1 coincide the coordinate axes at A(−1, 0) and B(0, 1) respectively.After connecting these points we will get the line x

_{1}− x_{2}= −1.Thus,

(0,0) does not assures the in equation x

_{1}− x_{2}≤ −1 .Thus,

The area in the plane which does not have the origin shows the solution set of the in equation x

_{1}− x_{2}≤ −1.The area shown by −x

_{1}+ x_{2}≤ 0 or x_{1}≥ x_{2}:The line −x

_{1}+ x_{2}= 0 or x_{1}= x_{2}is the line passing through (0, 0).The area to the right of the line x

_{1}= x_{2}will assure the given in equation −x_{1}+ x_{2}≤ 0.If we mark a point (1, 3) to the left of the line x

_{1}= x_{2}.Here, 1≤3 which is not assuring the in equation x

_{1}≥ x_{2}.Hence,

The area to the right of the line x

_{1}= x_{2}will assure the given in equation −x_{1}+ x_{2}≤ 0.The area shown by x

_{1}≥ 0 and x_{2}≥ 0:Thus,

All the points in the first quadrant assure these in equations.

Thus,

The first quadrant is the area shown by the in equations x

_{1}≥ 0 and x_{2}≥ 0.The suitable area determined by the system of constraints, x

_{1}− x_{2}≤ −1, −x_{1}+ x_{2}≤ 0, x_{1}≥ 0, and x_{2}≥ 0, are as follows.Here we can see that the suitable area of the given LPP does not exist.

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