# Maximize the objective function P = 3x + 5y under these constraints: x + y ≤ 6, x - y ≥ 2, x ≥ 0, y ≥ 0

I’m so damn confused and couldn’t start even a sentence. I’ve a test tomorrow. pls help.

Go to main content# Maximize the objective function P = 3x + 5y under these constraints: x + y ≤ 6, x - y ≥ 2, x ≥ 0, y ≥ 0

I’m so damn confused and couldn’t start even a sentence. I’ve a test tomorrow. pls help.

Given objective function,

P = 3x + 5y

Given constraints,

x + y ≤ 6, x - y ≥ 2, x ≥ 0, y ≥ 0

Writing inequalities as equation,

x + y = 6 --- (i)

x - y = 2 --- (ii)

x = 0, y = 0 --- (iii)

From (i),

x + y = 6

x |
0 | 6 |
---|---|---|

y |
6 | 6 |

Taking origin (0, 0) as the testing point in inequality of equation (i),

0 + 0 ≤ 6 [True]

So, its region is towards the testing point (shown using arrow in the graph).

From (ii),

x - y = 2

x |
0 | 2 |
---|---|---|

y |
-2 | 0 |

Taking origin (0, 0) as the testing point in inequality of equation (ii),

0 + 0 ≥ 2 [False]

So, its region is away from the testing point.

From (iii),

x = 0 and y = 0 represents the line on x-axis and y-axis respectively. Their inequality x ≥ 0 and y ≥ 0 shows that they lie in positive direction.

Now, plotting in graph,

From graph,

A(2, 0), B(6, 0) and C(4, 2) represents the common region.

Under the objective function, P = 3x + 5y,

Point | x |
y |
P = 3x + 5y | Value | Remarks |
---|---|---|---|---|---|

A(2, 0) | 2 | 0 | 3(2) + 5(0) | 6 | Minimum |

B(6, 0) | 6 | 0 | 3(6) + 5(0) | 18 | |

C(4, 2) | 4 | 2 | 3(4) + 5(2) | 22 | Maximum |

Hence, the objective function P = 3x + 5y has maximum value 22 at C(4, 2).