# Solve system of equation in x.y and z

$\textbf{Solve system of equations:}\\\\$\mathbf{x+y-z=7}$\\\\$\mathbf{x^2+y^2-z^2=37}$\\\\$\mathbf{x^3+y^3-z^3=1}\$

36
rahul ·

maybe lengthy ... but i think i can try my hand at this... :P

x + y = 7 + z --- (i)

(x + y)2 - 2xy = 37 + z2

=> (7 + z)2 - 2xy = 37 + z2

=> 2xy = 49 + 14z + z2 - 37 - z2 => 2xy = 12 + 14z

=> xy = 6 + 7z ----- (ii)

Again, x3 + y3 = 1 + z3

=> (x + y)(x2 + y2 - xy) = (1 + z)(1 + z2 - z)

=> (7 + z)(37 + z2 - 6 - 7z) = (1 + z)(1 + z2 - z)

=> (7 + z)(31 + z2 - 7z) = (1 + z)(1 + z2 - z)

=> 217 + 7z2 - 49z + 31z + z3 - 7z2 = 1 + z2 - z + z + z3 - z2

=> 217 - 18z = 1 => 18z = 216 => z = 12

Thus, z = 12

Now its easy to find x and y....!!

71
Vivek @ Born this Way ·

Fine. But i guess there should be another neat solution also.