SAGE MATH

Single Variable (x)

\[x + 2x^2 + x^3 = 100\]

Using SageMath

Define X as variable in the domain of integer numbers and solve for x

ZZ (domains of integer numbers)

x =  var('x', domain=ZZ)
leq = x + 2*x**2 + x**3
sol = solve(leq == 100, x)
print(sol) # 4

To verify

\[4 + 2*(4^2) + (4)^3 = 100\]

we can confirm with

leq(x=4) # 100

solving

\[x_0^4 - 150x_0^3 + 4389x_0^2 - 43000x_0 +131100 = 0\]

sage

x =  var('x', domain=ZZ)
leq = x**4 - 150*x**3 + 4389*x**2 - 43000*x + 131100
sol = solve(leq == 0, x)
sol

Linear Equation

Two Variables (x, y)

\[x + y = 10\]

x = var('x', domain=ZZ)
y = var('y', domain=ZZ)
sol = solve(x+y==10, (x,y))
sol

Solution

\[x = t_0, y = -t_0 + 10\]

Say we have 2 equations

\[x + y = 10, x=y\]

Solution

x = var('x', domain=ZZ)
y = var('y', domain=ZZ)
sol = solve([x+y==10, x==y], (x,y))
sol

Three Variables (x, y, z)

\[\begin{cases} 2x + y = 15 \\ x + y + z = 20 \\ 3z = 30 \end{cases}\]

Solution

x = var('x', domain=ZZ)
y = var('y', domain=ZZ)
z = var('z', domain=ZZ)
sol = solve([x + x + y == 15, z + z + z==30, x+y+z ==20], (x,y,z))
sol

Matrix

Inverse of matrix Sage math

\[\begin{bmatrix} 0 & 2 & 0 & 0 \\ 3 & 0 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 7 \\ \end{bmatrix}\]

A = matrix([[0,2,0,0], [3,0,0,0], [0,0,5,0], [0,0,0,7]])
A.inverse()

Fields

Finite Field of size 7

Zp = GF(7)