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in Mathematics by (60.8k points)
Using the method of integration, find the area of the region bounded by the following lines: 3x - y - 3 = 0, 2x + y - 12 = 0, x - 2y - 1 = 0

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Given lines are
3x - y - 3 = 0 ...(i)
2x + y - 12 = 0 ...(ii)
x - 2y - 1 = 0 ...(iii)
For intersecting point of (i) and (ii)
(i) + (ii) => 3x - y - 3 + 2x + y - 12 = 0
=> 5x - 15 = 0
=> x = 3
Putting x = 3 in (i), we get
9 - y - 3 = 0
=> y = 6
Intersecting point of (i) and (ii) is (3, 6)
For intersecting point of (ii) and (iii)
(ii) – 2 × (iii) => 2x + y - 12 - 2x + 4y + 2 = 0
=> 5y - 10 = 0
=> y = 2
Putting y = 2 in (ii) we get
2x + 2 - 12 = 0
=> x = 5
Intersecting point of (ii) and (iii) is (5, 2).
For Intersecting point of (i) and (iii)
(i) – 3 × (iii) => 3x - y - 3 - 3x + 6y + 3 = 0
=> 5y = 0
=> y = 0
Putting y = 0 in (i), we get
3x - 3 = 0
=> x = 1
Intersecting point (i) and (iii) is (1, 0).

Graph

Shaded region is required region.

Required Area

Finding the Area

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