StudentAidUnit 5: Linear Equations and Inequalities — Exercise 5 2
9th Class Mathematics · Unit 5: Linear Equations and Inequalities
5.2.1.Maximize f(x,y) = 2x + 5y; subject to the constraints 2y - x ≤ 8; x - y ≤ 4; x ≥ 0; y ≥ 0
Given
Constraints
-x+2yle 8;quad x-yle 4;quad xge0, yge0
Formula
Associated equations
-x+2y=8 (iii);quad x-y=4 (iv)
Working
Intercepts
(iii): (0,4), (-8,0)qquad (iv): (0,-4), (4,0)
Solve (iii)+(iv)
(-x+2y)+(x-y)=8+4 ⇒ y=12; put in (iii): x=16, D(16,12)
Corner points
A(0,0), B(4,0), C(0,4), D(16,12)
z(A)=2(0)+5(0)=0;quad z(B)=2(4)+5(0)=8;quad z(C)=2(0)+5(4)=20;quad z(D)=2(16)+5(12)=92
Result
Maximum
z=2x+5y is maximum at (16,12)
5.2.2.Maximize f(x,y) = x + 3y; subject to the constraints 2x + 5y ≤ 30; 5x + 4y ≤ 20; x ≥ 0; y ≥ 0
Given
Constraints
2x+5yle 30;quad 5x+4yle 20;quad xge0, yge0
Formula
Associated equations
2x+5y=30;quad 5x+4y=20
Working
Intercepts
2x+5y=30: (0,6),(15,0)qquad 5x+4y=20: (0,5),(4,0)
Corner points
A(0,0), B(4,0), C(0,5)
z(A)=0+0=0;quad z(B)=4+0=4;quad z(C)=0+15=15
Result
Maximum
z=x+3y is maximum at (0,5)
5.2.3.Maximize z = 2x + 3y; subject to the constraints: 2x + y ≤ 4; 4x - y ≤ 2; x ≥ 0; y ≥ 0
Given
Constraints
2x+yle4;quad 4x-yle2;quad xge0, yge0
Formula
Associated equations
2x+y=4;quad 4x-y=2
Working
Intercepts
2x+y=4: (0,4),(2,0)qquad 4x-y=2: (0,-2),left(tfrac12,0right)
Solve (iii)+(iv)
(2x+y)+(4x-y)=4+2 ⇒ x=1; put in (iii): y=2, P(1,2)
Corner points
(0,0), left(tfrac12,0right), (0,4), (1,2)
z(0,0)=0;quad zleft(tfrac12,0right)=1;quad z(0,4)=12
At P
z(1,2)=2(1)+3(2)=7
2(1)+3(2) = 2+6 = 8, not 7. This is an arithmetic slip; it does not change which corner gives the maximum.
Result
Maximum
z=2x+3y is maximum at (0,4)
5.2.4.Minimize z = 2x + y; subject to the constraints: x + y ≥ 3; 7x + 5y ≤ 35; x ≥ 0; y ≥ 0
Given
Constraints
x+yge3;quad 7x+5yle35;quad xge0, yge0
Formula
Associated equations
x+y=3;quad 7x+5y=35
Working
Intercepts
x+y=3: (0,3),(3,0)qquad 7x+5y=35: (0,7),(5,0)
Corner points
A(3,0), B(0,3), C(5,0), P(0,7)
z(A)=6;quad z(B)=3;quad z(C)=10;quad z(P)=7
Result
Minimum
z=2x+y is minimum at (0,3)
5.2.5.Maximize the function defined as f(x,y) = 2x + 3y subject to the constraints: 2x + y ≤ 10; x + 2y ≤ 14; x ≥ 0; y ≥ 0
Given
Constraints
2x+yle10;quad x+2yle14;quad xge0, yge0
Formula
Associated equations
2x+y=10;quad x+2y=14
Working
Intercepts
2x+y=10: (0,10),(5,0)qquad x+2y=14: (0,7),(14,0)
Solve 2(iii)-(iv)
(4x+2y)-(x+2y)=20-14 ⇒ 3x=6 ⇒ x=2; put in (iii): y=6, C(2,6)
Corner points
A(0,0), B(5,0), C(2,6), D(0,7)
z(A)=0;quad z(B)=10;quad z(C)=2(2)+3(6)=22;quad z(D)=2(0)+3(7)=21
Result
Maximum
z=2x+3y is maximum at (2,6)
5.2.6.Find minimum and maximum values of z = 3x + y; subject to the constraints: 3x + 5y ≥ 15; x + 3y ≤ 9; x ≥ 0; y ≥ 0
Given
Constraints
3x+5yge15;quad x+3yle9;quad xge0, yge0
Formula
Associated equations
3x+5y=15;quad x+3y=9
Working
Intercepts
3x+5y=15: (0,3),(5,0)qquad x+3y=9: (0,3),(9,0)
The book's 'To find Points' line states putting x=0 into x+3y=9 gives y=2 and point (0,2); actually x=0 gives 3y=9, y=3, point (0,3) (which coincides with the y-intercept of the other line). The graph and the final corner-point list correctly use (0,3), so the error is confined to this one transcribed line.
Corner points
(0,3), (5,0), (9,0)
z(0,3)=0+3=3;quad z(5,0)=15+0=15;quad z(9,0)=27+0=27
Result
Minimum and Maximum
z=3x+y is minimum at (0,3) and maximum at (9,0)