Question
Solve the following systems of linear equations has no solution:
$\text{x}+2\text{y}\leq3,3\text{x}+4\text{y}\geq12,\text{x}\geq0,\text{y}\geq1.$
$\text{x}+2\text{y}\leq3,3\text{x}+4\text{y}\geq12,\text{x}\geq0,\text{y}\geq1.$
✓
Answer
Converting theinequations into equations, we get
$\text{x}+2\text{y}\leq3,3\text{x}+4\text{y}\geq12,\text{x}\geq0,\text{y}\geq1.$
Region represented by $\text{x}+2\text{y}\leq3:$
The line x + 2y = 3 meets the coordinate axes at $\Big(0,\frac{3}{2}\Big)$ and (3,0). We find that (0, 0) satisfies inequation $\text{x}+2\text{y}\leq3.$ So the portion containing origin represents the solution set of the inequation $\text{x}+2\text{y}\leq3.$
Region represented by $3\text{x}+4\text{y}\geq12:$
The line 3x + 4y = 12 meets the coordinate axes at (0, 3) and (4, 0). We find that (0, 0) does not satisfy inequation $3\text{x}+4\text{y}\geq12.$So the portion not containing the origin is represented by the inequation $3\text{x}+4\text{y}\geq12.$
Region represented by $\text{x}\geq0:$
Clearly, $\text{x}\geq0$ represents the region lying on the right side of y-axis.
Region represented by $\text{y}\geq1: $
The line y = 1 is parallel to x-axis. (0, 0) does not satisfy inequation $\text{y}\geq1.$ So the region lying above the line y = 1 is represented by $\text{y}\geq1.$
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