System of Equations

[nycmathguy]

Solve for x and y.

sqrt{x} + y = 7
x + sqrt{y} = 11

 

[MathLover1]

sqrt(x)+ y = 7..............1)
x + sqrt(y) = 11...........2)
-------------------------------------isolate sqrt
sqrt(x) = 7-y..............1)
sqrt(y) = 11-x...........2)
--------------------------------square both sides
x = (7-y)^2..............1)
y = (11-x)^2...........2)
----------------------------------------
x = y^2 - 14 y + 49..............1)
y = x^2 - 22 x + 121...........2)
-------------------------------------

substitute x from eq. 1)

y = (y^2 - 14 y + 49)^2 - 22 (y^2 - 14 y + 49) + 121...........2), solve for y

0= y^4 - 28 y^3 + 272 y^2 - 1064 y + 1444 -y

y^4 - 28 y^3 + 272 y^2 - 1065 y + 1444=0....factor

(y - 4) (y^3 - 24 y^2 + 176 y - 361) = 0

one solution: y=4

y^3 - 24 y^2 + 176 y - 361= 0 roots are
y≈3.4156
y≈9.8051
y≈10.779

we dont use approximate values

so, solution is x=9, y=4

intersection point: (9,4)

 

[nycmathguy]

sqrt(x)+ y = 7..............1)
x + sqrt(y) = 11...........2)
-------------------------------------isolate sqrt
sqrt(x) = 7-y..............1)
sqrt(y) = 11-x...........2)
--------------------------------square both sides
x = (7-y)^2..............1)
y = (11-x)^2...........2)
----------------------------------------
x = y^2 - 14 y + 49..............1)
y = x^2 - 22 x + 121...........2)
-------------------------------------

substitute x from eq. 1)

y = (y^2 - 14 y + 49)^2 - 22 (y^2 - 14 y + 49) + 121...........2), solve for y

0= y^4 - 28 y^3 + 272 y^2 - 1064 y + 1444 -y

y^4 - 28 y^3 + 272 y^2 - 1065 y + 1444=0....factor

(y - 4) (y^3 - 24 y^2 + 176 y - 361) = 0

one solution: y=4

y^3 - 24 y^2 + 176 y - 361= 0 roots are
y≈3.4156
y≈9.8051
y≈10.779

we dont use approximate values

so, solution is x=9, y=4

intersection point: (9,4)

I will not post anything crazy like this moving forward.