2/x+3/y=9/xy,4/x+9/y=21/xy.what is the value of x,y

2/x+3/y = 9/xy can be rewritten

2y + 3x = 9, or
4y + 6x = 18

4/x+9/y = 21/xy can be rewritten
4y + 9x = 21

Now you can eliminate the y unknown:

3x = 3
x = 1
Then solve for y with any equation in x and y.
4y + 6 = 18
4y = 12
y = 3

To find the values of x and y, we can solve the system of equations:

2/x + 3/y = 9/xy ----(1)
4/x + 9/y = 21/xy ----(2)

Step 1: Multiply both sides of equation (1) by xy to eliminate the denominators:
(2 * xy / x) + (3 * xy / y) = 9
2y + 3x = 9xy ----(3)

Step 2: Multiply both sides of equation (2) by xy to eliminate the denominators:
(4 * xy / x) + (9 * xy / y) = 21
4y + 9x = 21xy ----(4)

Step 3: Now, we have a system of linear equations:
2y + 3x = 9xy ----(3)
4y + 9x = 21xy ----(4)

Step 4: Multiply equation (3) by 2 and equation (4) by -1 to make the coefficients of x the same:
4y + 6x = 18xy ----(5)
-4y - 9x = -21xy ----(6)

Step 5: Add equations (5) and (6) to eliminate y:
4y + 6x - 4y - 9x = 18xy - 21xy
-3x = -3xy

Step 6: Divide both sides of the equation by -3y:
x = y

Step 7: Substitute the value of x back into equation (3):
2y + 3(y) = 9y(y)
2y + 3y = 9y^2
5y = 9y^2

Step 8: Divide both sides of the equation by y:
5 = 9y

Step 9: Divide both sides by 9:
5/9 = y

Step 10: Substitute the value of y back into equation (6):
-4y - 9(x) = -21(x)(y)
-4(5/9) - 9x = -21(x)(5/9)
-20/9 - 9x = -35x/3

Step 11: Multiply both sides by 9 to eliminate the fraction:
-20 - 81x = -105x

Step 12: Add 105x to both sides:
105x - 81x = 20
24x = 20

Step 13: Divide both sides by 24:
x = 20/24
x = 5/6

So the values of x and y that satisfy both equations are x = 5/6 and y = 5/9.