Find the largest possible value of x^3+y^3+z^3 for realx, y, z, such that
xyz^2=−64y−128x
x^2yz=−32y−32z
3xy^2z=128x−64z
My answer is 440
My way of solving:
xyz(z)=−64y−128x (1)
xyz(x)=−32y−32z (2)
3xyz(y)=128x−64z (3)
(1)-(2)
xyz(z-2x)=64(z-2x)
xyz=64 ,z=2x
if z=2x,x=y=z=0
if xyz=64,
sub xyz=64 into (1),(2),(3)
simplify it and you can get:
z=-y-2x (4)
2x=-y-z (5)
3y=2x-z (6)
(5)=(6)because xyz=64 is derived from (1)and(2)
equate it and you will get:
y=2x,z=-4x
x^3+y^3+z^3=-55x^3=440
thanks
To find the largest possible value of x^3 + y^3 + z^3, we need to solve the given system of equations:
xyz^2 = -64y - 128x ...(1)
x^2yz = -32y - 32z ...(2)
3xy^2z = 128x - 64z ...(3)
First, notice that we can rearrange equation (1) as follows:
xyz^2 + 128x + 64y = 0
Now let's focus on eliminating variables to simplify the equations. We can start by multiplying equation (2) by z and equation (3) by x, which gives us:
x^3yz = -32xy - 32z^2 ...(4)
3xy^3z = 128x^2 - 64xz ...(5)
Next, we can substitute equation (4) and equation (5) into equation (1) to eliminate some variables. This gives us:
(-32xy - 32z^2) + 128x + 64y = 0
Simplifying this equation, we get:
-32xy + 128x + 64y - 32z^2 = 0
Let's rearrange this equation to solve for x:
x(-32y + 128) = -64y + 32z^2
x = (-64y + 32z^2) / (128 - 32y) ...(6)
Now, substitute equation (6) into equations (4) and (5):
((-64y + 32z^2) / (128 - 32y))^3 * yz = -32xy - 32z^2 ...(7)
3(-64y + 32z^2) / (128 - 32y) * y^3z = 128((-64y + 32z^2) / (128 - 32y))^2 - 64xz ...(8)
Simplify equations (7) and (8) to obtain two new equations:
(-64y + 32z^2)^3 / (128 - 32y)^3 * yz = -32xy - 32z^2 ...(9)
384(-64y + 32z^2)^2 / (128 - 32y)^3 * y^2z = 128(-64y + 32z^2) / (128 - 32y) - 64xz ...(10)
Now we can solve equations (9) and (10) simultaneously to find the values of y and z. Unfortunately, these equations are quite involved algebraically, and finding exact solutions could be quite challenging.
Therefore, to find approximate or numerical solutions, you can use numerical methods such as iterative approximation, substitution, or MATLAB. These methods can help you solve the system of equations numerically and find the values of y and z.
Once you have determined the values of y and z, you can substitute them back into equation (6) to find the value of x.
Finally, you can calculate x^3 + y^3 + z^3 using the values of x, y, and z that you obtained.