What is the area of the region created by the two linear equations 2x + y = 6 and 2x - y = -2 bounded by these lines and the x-axis?

Given:

Two linear equations: 2x + y = 6 and 2x - y = -2

Formula used:

Area of a triangle = 1/2 × base × height

Calculation:

Finding the points of intersection with the x-axis:

For 2x + y = 6:

⇒ y = 0

⇒ 2x + 0 = 6

⇒ x = 3

Point: (3, 0)

For 2x - y = -2:

⇒ y = 0

⇒ 2x - 0 = -2

⇒ x = -1

Point: (-1, 0)

Finding the point of intersection of the two lines:

Adding the two equations:

2x + y + 2x - y = 6 - 2

⇒ 4x = 4

⇒ x = 1

Substitute x = 1 into 2x + y = 6:

⇒ 2(1) + y = 6

⇒ 2 + y = 6

⇒ y = 4

Point: (1, 4)

Now we have the vertices of the triangle: (3, 0), (-1, 0), and (1, 4)

Base (distance between (-1, 0) and (3, 0)):

⇒ 3 - (-1) = 4

Height (y-coordinate of (1, 4)):

⇒ 4

Area:

⇒ 1/2 × base × height

⇒ 1/2 × 4 × 4

⇒ 8 sq. units

∴ The correct answer is option 2.