1227.sagews

EXAMPLE 1  Solve:

$x^2 = 2y + 10$

$3x - y = 9$

var('x y')

solve([x^2 == 2*y + 10, 3*x - y == 9], [x, y])

f(x) = 3*x - 9
g(x) = (x^2 - 10)/2

show(plot(f, -4, 5)+plot(g, -4, 5), ymin = -9, aspect_ratio = 1)

EXAMPLE 2  Solve:

$x - y = 3$

$(x - 2)^2 + (y + 3)^2 = 4$

var('x y')

solve([x-y == 3, (x-2)^2 + (y+3)^2 == 4], [x,y])

f(x) = x - 3
g(x) = sqrt(4 - (x-2)^2) - 3
h(x) = -sqrt(4 - (x-2)^2) - 3

show(plot(f, -1, 4) + plot(g, 0, 4)+plot(h, 0, 4), aspect_ratio = 1)

EXAMPLE 3  Solve:

$4x^2 + y^2 = 13$

$x^2 + y^2 = 10$

var('x y')

solve([4*x^2 + y^2 == 13, x^2 + y^2 == 10], [x, y])

f1(x) = sqrt(13 - 4*x^2)
f2(x) = -sqrt(13 - 4*x^2)
g1(x) = sqrt(10 - x^2)
g2(x) = -sqrt(10 - x^2)

show(plot(f1, -sqrt(13)/2, sqrt(13)/2) + plot(f2, -sqrt(13)/2, sqrt(13)/2) + plot(g1, -sqrt(10), sqrt(10)) + plot(g2, -sqrt(10), sqrt(10)))

EXAMPLE 4  Solve:

$y = x^2 + 3$

$x^2 + y^2 = 9$

var('x y')

solve([y == x^2+3, x^2+y^2==9], [x, y])

f(x) = x^2 + 3
g1(x) = sqrt(9-x^2)
g2(x) = -sqrt(9-x^2)

show(plot(f, -3, 3)+plot(g1, -3, 3)+plot(g2, -3, 3), aspect_ratio = 1)

EXAMPLE 5

$2x + 2y + y = 36$

$xy = 54$

var('x y')

solve([2*x+3*y == 36, x*y == 54], [x, y])

f(x) = (36-2*x)/3
g(x) = 54/x

plot(f, 0, 20)+plot(g, 1, 54)