Linear programming (graphical) Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Linear programming (graphical) questions. See exactly how to solve problems on linear-programming, feasible-region, vertex-method, formulation.

linear-programmingfeasible-regionvertex-methodformulationconstraintsbinding-constraints
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
The feasible region RR is defined by the constraints x+2y≤10x+2y\le10, y≤4y\le4, x≥0x\ge0 and y≥0y\ge0. The point (2,4)\left(2,4\right) is a vertex of RR. Find the value of the objective function P=x+4yP=x+4y at this vertex.

Worked solution

  1. State the objective function

    P=x+4yP=x+4y

    The objective is the linear expression whose value is required.

  2. Confirm that (2,4)\left(2,4\right) is a vertex by checking the boundaries through it

    (2)+2(4)=10  (x+2y=10),(4)=4  (y=4)\left(2\right)+2\left(4\right)=10\;(x+2y=10),\quad \left(4\right)=4\;(y=4)

    Two boundary lines meet at this point and it satisfies every constraint.

  3. Substitute the coordinates of the vertex

    P=1(2)+4(4)P=1\left(2\right)+4\left(4\right)

    The objective is evaluated at a point by substituting its coordinates.

  4. Work out the value

    P=2+16=18P=2+16=18

    This is the value of the objective at this corner of RR.

Answer
P=18P=18
Question 2
2 markseasy
The feasible region RR is defined by the constraints 2x+3y≤402x+3y\le40, x≤11x\le11, x≥0x\ge0 and y≥0y\ge0. The point (11,6)\left(11,6\right) is a vertex of RR. Find the value of the objective function P=8x+3yP=8x+3y at this vertex.

Worked solution

  1. State the objective function

    P=8x+3yP=8x+3y

    The objective is the linear expression whose value is required.

  2. Confirm that (11,6)\left(11,6\right) is a vertex by checking the boundaries through it

    2(11)+3(6)=40  (2x+3y=40),(11)=11  (x=11)2\left(11\right)+3\left(6\right)=40\;(2x+3y=40),\quad \left(11\right)=11\;(x=11)

    Two boundary lines meet at this point and it satisfies every constraint.

  3. Substitute the coordinates of the vertex

    P=8(11)+3(6)P=8\left(11\right)+3\left(6\right)

    The objective is evaluated at a point by substituting its coordinates.

  4. Work out the value

    P=88+18=106P=88+18=106

    This is the value of the objective at this corner of RR.

Answer
P=106P=106
Question 3
2 markseasy
The feasible region RR is defined by the constraints 4x+3y≤354x+3y\le35, y≤9y\le9, x≥0x\ge0 and y≥0y\ge0. The point (2,9)\left(2,9\right) is a vertex of RR. Find the value of the objective function P=x+5yP=x+5y at this vertex.

Worked solution

  1. State the objective function

    P=x+5yP=x+5y

    The objective is the linear expression whose value is required.

  2. Confirm that (2,9)\left(2,9\right) is a vertex by checking the boundaries through it

    4(2)+3(9)=35  (4x+3y=35),(9)=9  (y=9)4\left(2\right)+3\left(9\right)=35\;(4x+3y=35),\quad \left(9\right)=9\;(y=9)

    Two boundary lines meet at this point and it satisfies every constraint.

  3. Substitute the coordinates of the vertex

    P=1(2)+5(9)P=1\left(2\right)+5\left(9\right)

    The objective is evaluated at a point by substituting its coordinates.

  4. Work out the value

    P=2+45=47P=2+45=47

    This is the value of the objective at this corner of RR.

Answer
P=47P=47
Question 4
2 markseasy
The feasible region RR is defined by the constraints x+2y≤19x+2y\le19, y≤6y\le6, x≥0x\ge0 and y≥0y\ge0. The point (7,6)\left(7,6\right) is a vertex of RR. Find the value of the objective function P=x+3yP=x+3y at this vertex.

Worked solution

  1. State the objective function

    P=x+3yP=x+3y

    The objective is the linear expression whose value is required.

  2. Confirm that (7,6)\left(7,6\right) is a vertex by checking the boundaries through it

    (7)+2(6)=19  (x+2y=19),(6)=6  (y=6)\left(7\right)+2\left(6\right)=19\;(x+2y=19),\quad \left(6\right)=6\;(y=6)

    Two boundary lines meet at this point and it satisfies every constraint.

  3. Substitute the coordinates of the vertex

    P=1(7)+3(6)P=1\left(7\right)+3\left(6\right)

    The objective is evaluated at a point by substituting its coordinates.

  4. Work out the value

    P=7+18=25P=7+18=25

    This is the value of the objective at this corner of RR.

Answer
P=25P=25
Question 5
2 markseasy
The feasible region RR is defined by the constraints x+2y≤39x+2y\le39, y≤3y\le3, x≥0x\ge0 and y≥0y\ge0. The point (33,3)\left(33,3\right) is a vertex of RR. Find the value of the objective function P=2x+6yP=2x+6y at this vertex.

Worked solution

  1. State the objective function

    P=2x+6yP=2x+6y

    The objective is the linear expression whose value is required.

  2. Confirm that (33,3)\left(33,3\right) is a vertex by checking the boundaries through it

    (33)+2(3)=39  (x+2y=39),(3)=3  (y=3)\left(33\right)+2\left(3\right)=39\;(x+2y=39),\quad \left(3\right)=3\;(y=3)

    Two boundary lines meet at this point and it satisfies every constraint.

  3. Substitute the coordinates of the vertex

    P=2(33)+6(3)P=2\left(33\right)+6\left(3\right)

    The objective is evaluated at a point by substituting its coordinates.

  4. Work out the value

    P=66+18=84P=66+18=84

    This is the value of the objective at this corner of RR.

Answer
P=84P=84

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