How do I solve systems of equations to find local max, min, and saddle points?

[says]

Homework Statement

Find the local max, min, and saddle point for the function:
f(x,y) = 2x^2+3xy+4y^2-5x+2y

Homework Equations

The Attempt at a Solution

I've taken the two partial derivatives

Fx = 4x + 3y - 5
Fy = 3x + 8y + 2

I know that the critical points will sit where both of theses partial derivatives = 0
i.e.

Fx = 4x + 3y - 5 = 0
Fy = 3x + 8y + 2 = 0

The problem I have here though is that I don't know how to solve the system of equations.

I know once I've solved the system of equations I can use the determinant of the jacobian matrix to see whether they are local max, min, or saddle points...

Any help with solving the system of equations would be much appreciated. I've had a bit of trouble solving systems of equations in the past.

 

[SteamKing]

Homework Statement

Find the local max, min, and saddle point for the function:
f(x,y) = 2x^2+3xy+4y^2-5x+2y

Homework Equations

The Attempt at a Solution

I've taken the two partial derivatives

Fx = 4x + 3y - 5
Fy = 3x + 8y + 2

I know that the critical points will sit where both of theses partial derivatives = 0
i.e.

Fx = 4x + 3y - 5 = 0
Fy = 3x + 8y + 2 = 0

The problem I have here though is that I don't know how to solve the system of equations.

Really? You never solved a set of simultaneous linear equations in your algebra courses?

You can use Cramer's Rule or elimination to solve the system above.

http://www.coolmath.com/algebra/14-determinants-cramers-rule/01-determinants-cramers-rule-2x2-01

http://www.purplemath.com/modules/systlin6.htm

I know once I've solved the system of equations I can use the determinant of the jacobian matrix to see whether they are local max, min, or saddle points...

Any help with solving the system of equations would be much appreciated. I've had a bit of trouble solving systems of equations in the past.

 

[HallsofIvy]

Generally speaking, it is not a good idea to try to learn Calculus until after you have a firm grasp of algebra. You have the equations
Fx = 4x + 3y - 5 = 0 and Fy = 3x + 8y + 2 = 0. If you multiply the first equation by 3 you have 12x+ 9y- 15= 0. If you multiply the second equation by 4 you have 12x+ 32y+ 8= 0. Now the x term in each equation has the same coefficient so subtracting one equation from the other eliminates x and you have a single equation to solve for y.