Solution of System of Equations: [Lambda] = 3/(2x)

[snoggerT]

[lambda]=3/(2x) = 1/(2y) = 1/(3z) with constraint eq. x^2+2y^2+6z^2=1

The Attempt at a Solution

- I can't figure out how to solve this system of equations. Can somebody please help.

 

[Defennder]

Is $$\lambda$$ supposed to be constant? If so, then it appears simple. Just Express x,y,z in terms of lambda, substitute it into the constraint equation and solve for lambda.

 

[snoggerT]

Is $$\lambda$$ supposed to be constant? If so, then it appears simple. Just Express x,y,z in terms of lambda, substitute it into the constraint equation and solve for lambda.

- yeah, that's way simpler than I was making it. So I would take lambda and plug back into find my x, y, z, right?

 

[Defennder]

Yes that's right.

 

[snoggerT]

Yes that's right.

- so I get lambda to be +/- sqrt(41/12), which can be reduced to +/- 1/2*sqrt(41/3), but that gets somewhat ugly when plugging back into find my x,y,z. I'm trying to find the min/max on this problem, so I have to find my x,y,z to plug back into the f(x,y,z) function. Is there any way to simplify things further?

 

[rocomath]

Wait wait! You're solving for lambda?

$$\lambda=\frac{3}{2x}$$

$$\lambda=\frac{1}{2y}$$

$$\lambda=\frac{1}{3z}$$

Set them equal to each other so that you have y = ... in terms of x and z = ... in terms of x so you can plug it back into your equation to solve for x.

 

[snoggerT]

Wait wait! You're solving for lambda?

$$\lambda=\frac{3}{2x}$$

$$\lambda=\frac{1}{2y}$$

$$\lambda=\frac{1}{3z}$$

Set them equal to each other so that you have y = ... in terms of x and z = ... in terms of x so you can plug it back into your equation to solve for x.

- so I get +/- sqrt(27/41) for x, is that right?

 

[rocomath]

- so I get +/- sqrt(27/41) for x, is that right?

Hmm, what did you do?

 

[snoggerT]

Hmm, what did you do?

- I set 3/(2x)= 1/(2y) and got y in terms of x and did the same for z. then plugged into the constraint equation and solved for x. I'm guessing I did it wrong though.

 

[rocomath]

$$y=\frac x 3$$

$$z=\frac 2 9 x$$

 

[snoggerT]

$$y=\frac x 3$$

$$z=\frac 2 9 x$$

- that's what I got for y and z, so I must of done something wrong with my algebra in the constraint equation. Let me work it again.

 

[snoggerT]

I still get x=+/- sqrt(27/41). So I'm not sure what I'm doing wrong. This is how I'm setting the problem up:

x^2+2(x/3)^2+6(2x/9)^2=1

 

[Dick]

This is a lagrange multiplier problem isn't it? Can you state the original problem?

 

[snoggerT]

This is a lagrange multiplier problem isn't it? Can you state the original problem?

- sure. I probably need to start over anyways because I think I'm just confusing myself.

find the min/max:

f(x,y,z)=3x+2y+4z with constraint equation x^2+2y^2+6z^2=1

 

[Dick]

Ok, I think you are on completely the right track. lambda^2=12/41 is what I get too. And yes, x=+/-sqrt(27/41). I'll admit the numbers aren't that pretty. But I think you are doing exactly the right thing. Just keep on doing it.