Given Equations:

1/16x + 1/15y = 9/20 ...(1)
1/20x - 1/27y = 4/45 ...(2)

Step 1: Simplify the Equations

To solve the given system of equations, we first simplify them by finding the common denominators.

Multiplying equation (1) by 16 and equation (2) by 20, we get:

x/1 + (16/15)y = (16/15)(9/20) ...(3)
(20/20)x - (20/27)y = (20/27)(4/45) ...(4)

Simplifying further:

x + (32/30)y = (144/300)
x - (20/27)y = (80/1215)

Step 2: Eliminate x by multiplying equations

To eliminate x, we multiply equation (3) by 27 and equation (4) by 30:

27x + (432/30)y = (144/300)(27) ...(5)
30x - (20/27)(30)y = (80/1215)(30) ...(6)

Simplifying further:

27x + 14.4y = 12.96 ...(5)
30x - 22.22y = 0.533 ...(6)

Step 3: Eliminate x by subtracting equations

To eliminate x, we subtract equation (6) from equation (5):

(27x - 30x) + (14.4y + 22.22y) = 12.96 - 0.533
-3x + 36.62y = 12.427 ...(7)

Step 4: Solve for y

From equation (7), we can solve for y:

36.62y = 12.427 + 3x
y = (12.427 + 3x)/36.62 ...(8)

Step 5: Substitute y into one of the original equations

Substituting equation (8) into equation (1), we get:

1/16x + 1/15[(12.427 + 3x)/36.62] = 9/20

Simplifying further:

1/16x + (12.427 + 3x)/548.7 = 9/20

Step 6: Solve for x

To solve for x, we can multiply through by 16 to eliminate the fraction:

x + (12.427 + 3x)/34.294 = 0.72

Simplifying further:

34.294x + 12.427 + 3x = 0.72(34.294)

37.294x + 12.427 = 24.707

37.294x = 24.707 - 12.427

37.294x = 12.28

x = 12.28/37.294

x ≈ 0.329

Step 7: Substitute x back into y equation

Substituting x = 0.329 into equation (8), we get:

y = (12.427 + 3(0.329