Help solving for 3 equations and 3 unknowns

[skybox]

Homework Statement

Solve the following equation for $P_{1}$ and $P_{2}$

Homework Equations

$7.0+0.004P_{1}-\lambda(1-0.0004P_{1})=0$
$7.0+0.004P_{2}-\lambda=0$
$P_{1}+P_{2}-500-0.0002P_{1}^{2}=0$

The Attempt at a Solution

I am having some issues on ways to solve this problem. I guess the main point I am stuck on is how do I even approach it?

I have three unknowns $P_{1}, P_{2}, \lambda$ and three equations, so I should be able to solve for $P_{1}, P_{2}$.

The example shows that the solutions are:
$P_{1}=178.882 , P_{2}=327.496$.

Any suggestions or tips would be greatly appreciated

 

[skiller]

My first suggestion would be to give some equations. You've given 3 expressions, which don't appear to equate to anything.

 

[skybox]

My first suggestion would be to give some equations. You've given 3 expressions, which don't appear to equate to anything.

Woops. Updated. All expressions equal 0

 

[skiller]

Woops. Updated. All expressions equal 0

That's better!

How are you with algebraic manipulation?

The first equation gives you $P_1$ in terms of $\lambda$, and the second gives $P_2$ in terms of $\lambda$.

Plug them both into the third equation and you find the value of $\lambda$.

Use this value of $\lambda$ to find the values of $P_1$ and $P_2$.

Job done!

 

[skybox]

That's better!

How are you with algebraic manipulation?

The first equation gives you $P_1$ in terms of $\lambda$, and the second gives $P_2$ in terms of $\lambda$.

Plug them both into the third equation and you find the value of $\lambda$.

Use this value of $\lambda$ to find the values of $P_1$ and $P_2$.

Job done!

Thanks for the help. This is becoming a very complicated solution :S Will just use Matlab to solve!