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Linear Inequations — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLinear Inequations5 Marks
Question
Solve the following systems of linear inequations graphically:
$\text{x}+2\text{y}\leq3,3\text{x}+4\text{y}\geq12,\text{y}\geq1,\text{x}\geq0,\text{y}\geq0$
✓

Answer



We have,

$\text{x}+2\text{y}\leq3,3\text{x}+4\text{y}\geq12,\text{y}\geq1,\text{x}\geq0,\text{y}\geq0$

Converting the inequations into equations, we get

x + 2y = 3, 3x + 4y = 12

y = 1, x = 0 and y = 0

Region represented by $\text{x}+2\text{y}\leq3$

Putting x = 0 in x + 2y = 3, we get $\text{y}=\frac{3}{2}$

Putting y = 0 in x + 2y = 3, we get $\text{x}=3$

$\therefore$ The line x + 2y = 3 meets the coordinate axes at $\Big(0,\frac{3}{2}\Big)$ and (4, 0) joining these point by a thick line.

Now, putting x = 0 and y = 0 in $3\text{x}+4\text{y}\leq3$ we get $0\leq12$

since, (0, 0) does not satisfies the inequality $3\text{x}+4\text{y}\leq3$ So, the portion containing the origin represents the solution $3\text{x}+4\text{y}\leq3$

Region represented by $\text{y}\geq1$ Clearly, y = 1 is a line parallel to x-axis at a distance 3 units from the origin. Since (0, 0) does not satisfies the inequation $\text{y}\geq1$

So, the portion not containing the origin is represented by the inequation

Region represented by $\text{x}\geq0$ and $\text{y}\geq\alpha$

Clearly, $\text{x}\geq0$ and $\text{y}\geq0$ represent the first quadrant.

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