1946.sagews

var('y')
MS = MatrixSpace(SR,2)
x_prime(x,y) =  x * ( 80 -x -y )
y_prime(x,y) = y * ( 120 -x-3*y -2*y^2)
f(x,y) = ( 80 -x -y )
g(x,y) = ( 120 -x-3*y -2*y^2)
fx(x,y) = derivative(x_prime(x,y),x)
fy(x,y) = derivative(x_prime(x,y),y)
gx(x,y) = derivative(y_prime(x,y),x)
gy(x,y) = derivative(y_prime(x,y),y)

def comm(x,y):
    A = x * fx(x,y) + f(x,y)
    B = x * fy(x,y)
    C = y * gx(x,y)
    D = y * gy(x,y) + g(x,y)
    return MS.matrix([A,B,C,D])

latex(comm(x,y))

\left(\begin{array}{rr} -{\left(2 \, x + y - 80\right)} x - x - y + 80 & -x^{2} \\ -y^{2} & -{\left({\left(4 \, y + 3\right)} y + 2 \, y^{2} + x + 3 \, y - 120\right)} y - 2 \, y^{2} - x - 3 \, y + 120 \end{array}\right)

solve([x_prime(x,y)==0,y_prime==0],x,y)

[[x == 0, y == 0], [x == 80, y == 0], [x == 0, y == -1/4*sqrt(969) - 3/4], [x == 0, y == 1/4*sqrt(969) - 3/4], [x == 76, y == 4], [x == 85, y == -5]]

latex(solve([x_prime(x,y)==0,y_prime==0],x,y))

\left[\left[x = 0, y = 0\right], \left[x = 80, y = 0\right], \left[x = 0, y = -\frac{1}{4} \, \sqrt{969} - \frac{3}{4}\right], \left[x = 0, y = \frac{1}{4} \, \sqrt{969} - \frac{3}{4}\right], \left[x = 76, y = 4\right], \left[x = 85, y = \left(-5\right)\right]\right]

latex(comm(0,0))

\left(\begin{array}{rr} 80 & 0 \\ 0 & 120 \end{array}\right)

latex(comm(80,0))

\left(\begin{array}{rr} -6400 & -6400 \\ 0 & 40 \end{array}\right)

latex(comm(0,1/4*sqrt(969) - 3/4))

\left(\begin{array}{rr} -\frac{1}{4} \, \sqrt{969} + \frac{323}{4} & 0 \\ -\frac{1}{16} \, {\left(\sqrt{969} - 3\right)}^{2} & -\frac{1}{8} \, {\left(\sqrt{969} - 3\right)}^{2} - \frac{1}{32} \, {\left(\sqrt{969} - 3\right)} {\left({\left(\sqrt{969} - 3\right)}^{2} + 2 \, {\left(\sqrt{969} - 3\right)} \sqrt{969} + 6 \, \sqrt{969} - 978\right)} - \frac{3}{4} \, \sqrt{969} + \frac{489}{4} \end{array}\right)

comm(0,-1/4*sqrt(969) - 3/4)

[ 1/4*sqrt(969) + 323/4 0] [ -1/16*(sqrt(969) + 3)^2 -1/8*(sqrt(969) + 3)^2 + 1/32*(sqrt(969) + 3)*((sqrt(969) + 3)^2 + 2*(sqrt(969) + 3)*sqrt(969) - 6*sqrt(969) - 978) + 3/4*sqrt(969) + 489/4]

latex(comm(76,4))

\left(\begin{array}{rr} -5776 & -5776 \\ -16 & -304 \end{array}\right)

comm(85,-5)

[-7225 -7225] [ -25 425]

line1(x) = 1*x + (80/76)
line2(x) = (120/76) + (4/76)*x
A = plot(line1(x),(0,(120/76)))
B = plot(line2(x),(0,(120/76)))
A+B