Solve the system of equations please?

enter image source here

2 Answers
Feb 20, 2018

See below.

Explanation:

Making #y = lambda x#

#{(1+4lambda^2=5 lambda),(x^2(2-lambda^2)=31):}#

or

#((lambda = 1/4, x = -4),(lambda = 1/4, x = 4),(lambda = 1, x = -sqrt[31]),(lambda = 1, x = sqrt[31]))#

and then

#((y = -1, x = -4),(y = 1, x = 4),(y = -sqrt(31), x = -sqrt[31]),(y= sqrt(31), x = sqrt[31]))#

Feb 20, 2018

#(2,-44/31)#, #(-2,44/31)#, #(sqrt31,sqrt31)#, #(-sqrt31,-sqrt31)#

Explanation:

from equation (1) we have
#x^2+4y^2=5xy# .............................(3)
now multiply equation (2) by 4, i.e
#8x^2-4y^2=124# .............................(4)
now adding equation (3) and (4), we get
#9x^2=5xy+124#
#9x^2-124=5xy#
#(9x^2-124)/"5x"=y# ................................(5)
now substitute equation (5) in equation 2 and by solving , we get
#x^4-47x^2+496=0# ................................(6)
solving equation (6) we get
#x=2, -2, sqrt31, -sqrt31#
now using these values in equation (6), we get
#y=-44/5, 44/5, sqrt31, -sqrt31# respectively.