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Linear programming — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsLinear programming5 Marks
Question
Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below:
$2\text{x}+4\text{y}\leq8$
$3\text{x}+\text{y}\leq6$
$\text{x}+\text{y}\leq4$
$\text{x}\geq0,\text{y}\geq0$
✓

Answer

Converting the inequations into equations, ew obtain the lines.

2x + 4y = 8, 3x + y = 6, x + y =4, x = 0, y = 0.

These lines are drawn on a suitable scale and the feasible region of the LPP is shaded in the graph.



From the graph we can see the corner point as (0, 2) and (2, 0).

Now, solving the equations 3x + y = 6 and 2x +4y = 8 we get the values of x and y as $\text{x}=\frac{8}{5}$ and $\text{y}=\frac{6}{5}$.

Substituting $\text{x}=\frac{8}{5}$ and $\text{y}=\frac{6}{5}$ in Z = 2x + 5y we get,

$\text{z}=2\Big(\frac{8}{5}\Big)+5\Big(\frac{6}{5}\Big)$

$\text{z}=\frac{46}{5}$

Hence maximum value of Z is $\frac{46}{5}$ at $\text{x}=\frac{8}{5}$ and $\text{y}=\frac{6}{5}$.

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