SOLUTION: Minimize C=2x+3y +4z subject to minimize the problem: 4x+2y+z=10 x+y-z=5 x≥0,y≥0,z≥0

    Minimize  C = 2x + 3y + 4z subject to restrictions

        4x + 2y + z = 10
         x +  y - z =  5
        x ≥ 0, y ≥ 0, z ≥ 0

The idea of the solution is THIS :


    1)  Two restrictions are equations of planes in 3D space.

        These planes are not parallel -- hence, they have non-empty intersection.

        Their intersection is a straight line, and the first step is to present this straight line in a parametric form

            x = f(z),  y = g(z)

        of the coordinate z.



     2)  Having it, we can present the objective function  C = 2x + 3y + 4z  as a linear function of only one single variable z


             C = C(x,y,z) = C(z) = 2*f(z) + 3*g(z) + z.


        Then it should be easy to find the minimum of the function C(z).

To express x as a function of z, write the given restrictions in the form

        4x + 2y = 10 -  z    (1)
         x +  y =  5 +  z    (2)


Eliminate "y". For it, multiply eq(2) by 2 (both sides). Keep eq(1) as is. You will get

        4x + 2y = 10 -  z    (3)
        2x + 2y = 10 + 2z    (4)
       

Subtract eq(4) from eq(3). You will get

        2x =     = -3z,     or    x = -1.5z.    


Thus I got "x" as this function of z :  x = f(z) = -1.5z.    (5)



To express y as a function of z, use the restrictions in the form (1), (2) again.


Eliminate "x". For it, multiply eq(2) by 4 (both sides). Keep eq(1) as is. You will get

        4x + 2y = 10 -  z    (6)
        4x + 4y = 20 + 4z    (7)
       

Subtract eq(6) from eq(7). You will get

        2y =     = 10 + 5z,     or    y = 2.5z + 5.    


Thus I got "y" as this function  of z :  y = g(z) = 2.5z + 5.    (8)


Now the objective function  C = 2x + 3y + 4z  on the straigh line  (5), (8) becomes

    C = 2*(-1.5z) + 3*(2.5z + 5) + 4z = -3z + 7.5z + 15 + 4z = 8.5z + 15.


In the first octant, this function INCREAZES as z increases;

so, it it OBVIOUS that the objective function on the straight line gets the minimum at z= 0, and this minimum iz equal to 8.5*0 + 15 = 15.


As the last step, we can determine the x- and y- coordinates of the minimal point.

They are  x = -1.5*0 = 0  and  y = 2.5*0 + 5 = 5.


ANSWER.  Under given restrictions, the objective function gets it minimum at the point (x,y,z) = (0,5,0),

         and the minimum value is 15.