The solution of the following system of equations:

2(ax - by) + a + 4b = 0, 2(bx + ay) + b - 4a = 0, is also the solution of the equation:

​Calculation:

Expand the given equations:

1) 2ax - 2by + a + 4b = 0

2) 2bx + 2ay + b - 4a = 0

Multiply equation (1) by 'a' and equation (2) by 'b':

a(2ax - 2by + a + 4b) = 0 ⇒ 2a²x - 2aby + a² + 4ab = 0 (3)

b(2bx + 2ay + b - 4a) = 0 ⇒ 2b²x + 2aby + b² - 4ab = 0 (4)

Add equations (3) and (4):

(2a²x - 2aby + a² + 4ab) + (2b²x + 2aby + b² - 4ab) = 0

⇒ 2a²x + 2b²x + a² + b² = 0

⇒ 2(a² + b²)x = -(a² + b²)

⇒ x = -1/2

Multiply equation (1) by 'b' and equation (2) by '-a':

b(2ax - 2by + a + 4b) = 0 ⇒ 2abx - 2b²y + ab + 4b² = 0 (5)

-a(2bx + 2ay + b - 4a) = 0 ⇒ -2abx - 2a²y - ab + 4a² = 0 (6)

Add equations (5) and (6):

(2abx - 2b²y + ab + 4b²) + (-2abx - 2a²y - ab + 4a²) = 0

⇒ -2b²y - 2a²y + 4b² + 4a² = 0

⇒ -2(a² + b²)y = -4(a² + b²)

⇒ y = 2

Now substitute x = -1/2 and y = 2 in the given options:

Option 1: 2x - 3y = 6 ⇒ 2(-1/2) - 3(2) = -1 - 6 = -7 ≠ 6

Option 2: 3x + 4y = 59 ⇒ 3(-1/2) + 4(2) = -3/2 + 8 = -1.5 + 8 = 6.5 ≠ 59

Option 3: 3x - 2y = 5 ⇒ 3(-1/2) - 2(2) = -3/2 - 4 = -1.5 - 4 = -5.5 ≠ 5

Option 4: 4x + 3y = 4 ⇒ 4(-1/2) + 3(2) = -2 + 6 = 4

The solution of the given system of equations is also the solution of the equation 4x + 3y = 4.