Using the method of integration, find the area of the region bounded by the following lines: 5x - y = 0, x + y = 0, and 2x - y = 0

Given lines are
5x - y = 0 — (1)
x + y = 0 — (2)
2x - y = 0 — (3)
For intersecting point of (1) and (2)
(1) + (2) × 2 ⇒ 5x - y + 2x + 2y = 0 ⇒ 7x + y = 0 ⇒ y = -7x
Putting x = 0 in (1), we get y = 0
Intersecting point of (1) and (2) is (0, 0)
For intersecting point of (1) and (3)
(1) × 2 - (3) × 5 ⇒ 10x - 2y - 10x + 5y = 0 ⇒ 3y = 0 ⇒ y = 0
Putting y = 0 in (1), we get x = 0
Intersecting point of (1) and (3) is (0, 0)
For intersecting point of (2) and (3)
2 × (2) - (3) ⇒ 2x + 2y - 2x + y = 0 ⇒ 3y = 0 ⇒ y = 0
Putting y = 0 in (2), we get x = 0
Intersecting point (2) and (3) is (0, 0)
The lines are concurrent at (0,0).
Let's find another point on each line.
Line 1: 5x - y = 0; If x = 1, y = 5. Point (1,5)
Line 2: x + y = 0; If x = 1, y = -1. Point (1,-1)
Line 3: 2x - y = 0; If x = 1, y = 2. Point (1,2)
Solving the equations simultaneously:
5x - y = 0
x + y = 0 => y = -x
Substitute into the first equation:
5x - (-x) = 0
6x = 0
x = 0
y = 0
Intersection point (0,0)
5x - y = 0
2x - y = 0
Subtracting the equations gives 3x = 0, so x = 0 and y = 0
Intersection point (0,0)
x + y = 0
2x - y = 0
Adding the equations gives 3x = 0, so x = 0 and y = 0
Intersection point (0,0)
All three lines intersect at (0,0). The area enclosed is 0.