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Question
Solve the following systems of linear inequations graphically:
x + y ≥ 1, 7x + 9y ≤ 63, x ≤ 6, y ≤ 5, x ≥ 0, y ≥ 0
Solution
Converting the inequations to equations, we obtain:
x + y =1, 7x + 9y = 63, x = 6, y = 5
x + y=1: This line meets the x-axis at (1, 0) and the y-axis at (0, 1). Draw a thick line joining these points.
We see that the origin (0, 0) does not satisfy the inequation x + y \[\geq\]1 So, the portion not containing the origin represents the solution set of the inequation x + y \[\geq\]1
7x + 9y =63: This line meets the x-axis at (9, 0) and the y-axis at (0, 7). Draw a thick line joining these points.
We see that the origin (0, 0) satisfies the inequation 7x + 9y ≤ 63 So, the portion containing the origin represents the solution set of the inequation 7x + 9y ≤ 63
x = 6: This line is parallel to the x-axis at a distance 6 units from it.
We see that the origin (0, 0) satisfies the inequation x ≤ 6 So, the portion containing the origin represents the solution set of the inequation x ≤ 6
y = 5: This line is parallel to the y-axis at a distance 5 units from it.
We see that the origin (0,0) satisfies the inequation y ≤ 5 So, the portion containing the origin represents the solution set of the inequation
y ≤ 5
Clearly, x ≥ 0, y ≥ 0 represents the first quadrant.
Hence, the shaded region in the figure represents the solution set of the given set of inequations.
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