Question
Solve the following systems of linear inequations graphically: $2\text{x}+3\text{y}\leq35,\text{y}\geq3,\text{x}\geq2,\text{x}\geq0,\text{y}\geq0$
✓
Answer
We have,
$2\text{x}+3\text{y}\geq35,\text{y}\geq3,\text{x}\geq2,\text{x}\geq0\text{ and }\text{y}\geq0$
Converting the inequations into equations, we get
2x + 3y = 35, y = 3, X = 2, x = 0 and y = 0.
Region represented by $2\text{x}+3\text{y}\geq35$
Putting x = 0 in 2x + 3y = 35, we get $\text{y}=\frac{35}{3}$
Putting y = 0 in 2x + 3y = 35, we get $\text{x}=\frac{35}{2}$
$\therefore$ The line 2x + 3y = 35 meets the coordinate axes at $\Big(0,\frac{35}{3}\Big)$ and $\Big(\frac{35}{2},0\Big)$ joining these point by a thick line.
Now, putting x = 0 and y = 0 in $2\text{x}+3\text{y}\leq35$ we get $0\leq35.$
Clearly, (0, 0) satisfies the inequality $2\text{x}+3\text{y}\leq35$ So, the portion containing the origin represents the solution $2\text{x}+3\text{y}\leq35$
Region represented by $\text{y}\geq3$
Clearly, y = 3 is a line parallel to x-axis at a distance 3 units from the origin. Since (0, 0) does not satisfies the inequation $\text{y}\geq 3.$
So, the portion not containing the origin is represented by the $\text{y}\geq 3.$
Region represented by $\text{x}\geq2$ Clearly, x = 2 is a line parallel to y-axis at a distance of 2 units from the origin. Since (0, 0) does not satisfies the inequation $\text{x}\geq2$ so, the portion not containing the origin is represented by the given inequation.
Region represented by $\text{x}\geq0$ and $\text{y}\geq0$ dearly, $\text{x}\geq0$ and $\text{y}\geq0$ represent the first quadrant.
The common region of the above five regions represents the solution set of the given inequations as shown below.
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