Tìm x; y; z

To solve this problem, we can start by simplifying the given equations.

From the first equation, 4/x - 3 = 8/y - 6, we can add 3 to both sides to isolate the first fraction:

4/x = 8/y - 3

Next, we can find a common denominator for the fractions on the right side:

4/x = (8 - 3y)/y

To eliminate the fractions, we can cross-multiply:

4y = xy(8 - 3y)

Expanding the right side:

4y = 8xy - 3xy^2

Rearranging the equation:

3xy^2 - 8xy + 4y = 0

Now let's look at the second equation, 8/y - 6 = 20/7 - 15. We can simplify this equation by finding a common denominator:

8/y = (20 - 15y)/7

Cross-multiplying:

8y = 7(20 - 15y)

Expanding the right side:

8y = 140 - 105y

Rearranging the equation:

105y^2 - 8y + 140 = 0

Finally, we have the equation xyz = 640.

Now we have a system of three equations:

3xy^2 - 8xy + 4y = 0
105y^2 - 8y + 140 = 0
xyz = 640

To solve this system, we can use substitution or elimination methods. However, since the equations involve variables in different terms, substitution may be more complicated. Therefore, we will use the elimination method.

First, let's solve the second equation for y:

105y^2 - 8y + 140 = 0

Using the quadratic formula, we find:

y = (-(-8) ± √((-8)^2 - 4(105)(140))) / (2(105))

Simplifying:

y = (8 ± √(64 + 58800)) / 210

y = (8 ± √58864) / 210

y = (8 ± 242) / 210

y = (8 + 242) / 210 or y = (8 - 242) / 210

y = 250 / 210 or y = -234 / 210

Simplifying:

y = 5/3 or y = -13/3

Now let's substitute these values of y back into the first equation:

3xy^2 - 8xy + 4y = 0

For y = 5/3:

3x(5/3)^2 - 8x(5/3) + 4(5/3) = 0

Simplifying:

25x/3 - 40x/3 + 20/3 = 0

-15x/3 + 20/3 = 0

-5x + 20 = 0

-5x = -20

x = 4

For y = -13/3:

3x(-13/3)^2 - 8x(-13/3) + 4(-13/3) = 0

Simplifying:

169x/3 + 104x/3 - 52/3 = 0

273x/3 - 52/3 = 0

91x - 52 = 0

91x = 52

x = 52/91

Therefore, the solutions to the system of equations are:

x = 4, y = 5/3
x = 52/91, y = -13/3

To find z, we can substitute these values of x and y into the third equation:

xyz = 640

For x = 4 and y = 5/3:

4(5/3)z = 640

20z/3 = 640

20z = 1920

z = 1920/20

z = 96

For x = 52/91 and y = -13/3:

(52/91)(-13/3)z = 640

-676z/273 = 640

-676z = 640(273)

z = 640(273)/-676

z = -27360/676

Therefore, the solutions to the system of equations are:

x = 4, y = 5/3, z = 96
x = 52/91, y = -13/3, z = -27360/676