The maximum value of $z = 5x + 3y$, subjected to the conditions $3x + 5y \le 15, 5x + 2y \le 10, x, y \ge 0$ is
Updated On: Jul 5, 2022
$\frac{235}{19}$
$\frac{325}{19}$
$\frac{523}{19}$
$\frac{532}{19}$
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The Correct Option isA
Solution and Explanation
Given, inequalities are $3x + 5y \le 15, 5x + 2y \le 10, x, y \ge 0$
Also, given $z = 5x + 3y$
At point $A \left(2, 0\right)$$z = 5 ? 2 + 0 = 10$
At point $B\left(\frac{20}{19}, \frac{45}{19}\right)$,
$z = \frac{5\times20}{19}+\frac{3\times45}{19} = \frac{235}{19}$
At point $C \left(0, 3\right)$$z = 5 \left(0\right) + 3 ? 3 = 9$
Hence, maximum value of z is $\frac{235}{19}.$
Linear programming is a mathematical technique for increasing the efficiency and effectiveness of operations under specific constraints. The main determination of linear programming is to optimize or minimize a numerical value. It is built of linear functions with linear equations or inequalities restricting variables.
Characteristics of Linear Programming:
Decision Variables: This is the first step that will determine the output. It provides the final solution to the problem.
Constraints: The mathematical form in which drawbacks are expressed, regarding the resource.
Data: They are placeholders for known numbers to make writing complex models simple. They are constituted by upper-case letters.
Objective Functions: Mathematically, the objective function should be quantitatively defined.
Linearity: The function's relation between two or more variables must be straight. It indicates that the variable's degree is one.
Finiteness: Input and output numbers must be finite and infinite. The best solution is not possible if the function consists infinite components.
Non-negativity: The value of the variable should be either positive (+ve) or 0. It can't be a negative (-ve) number.
