Given lines are
3x - 2y + 1 = 0 ...(i)
2x + 3y - 21 = 0 ...(ii)
x - 5y + 9 = 0 ...(iii)
For intersection of (i) and (ii)
Applying (i) × 3 + (ii) × 2, we get
9x - 6y + 3 + 4x + 6y - 42 = 0
=> 13x - 39 = 0
=> x = 3
Putting it in (i), we get
9 - 2y + 1 = 0
=> 2y = 10 => y = 5
Intersection point of (i) and (ii) is (3, 5)
For intersection of (ii) and (iii)
Applying (ii) – (iii) × 2, we get
2x + 3y - 21 - 2x + 10y - 18 = 0
=> 13y - 39 = 0
=> y = 3
Putting y = 3 in (ii), we get
2x + 9 - 21 = 0
=> 2x - 12 = 0
=> x = 6
Intersection point of (ii) and (iii) is (6, 3)
For intersection of (i) and (iii)
Applying (i) – (iii) × 3, we get
3x - 2y + 1 - 3x + 15y - 27 = 0
=> 13y - 26 = 0 => y = 2
Putting y = 2 in (i), we get
3x - 4 + 1 = 0
=> x = 1
Intersection point of (i) and (iii) is (1, 2)
With the help of point of intersection we draw the graph of lines (i), (ii) and (iii)
Shaded region is required region.
Area of Required region