Algebra equations

Question

If x^2 + y^2 = 39 and xy = 7, find the values of x-y

Answer 1

x^2 + y^2 = 39

xy = 7

short way to solve this is subtract 2xy from both sides

x^2-2xy+y^2 = 39-2xy = 39-2(7)= 25

x^2-2xy+y^2 = 25

(x-y)^2 = 25

x-y = +/-5

check these solutions,

solve more tediously algebraically or just graph the 2 equations

y=7/x

x^2 +49/x^2 = 39

x^4 +49 =39x^2

x^4 -39x^2 +49 = 0

let z=x^2

z^2 -39z +49 =0

use the quadratic formula

z=39/2 +/-(1/2)sqr(39^2-4(49))

x= +/-sqr(z) =+/-1.14,+/-6.14

x-y = +/-5

or

solve by graphing the circle and rectangular hyperbola

use a graphing calculator

they are centered at the origin, circle has radius sqr39

they intersect in 4 points, 2 in quadrant I, 2 in quadrant III

(x,y) = (1.14, 6.14), (6.14,1.14) in quadrant I

and (-1.14, -6.14), (-6.14, 1.14) in quadrant III

or

x^2+y^2 = 39,

x^2 -2xy+y^2 = (x-y)^2 = 39 -2xy = 39-2(7)=39-14=25

x-y = +/-5

graph x-y=5 and x-y=-5

those 2 lines also intersect the circle and hyperbola in the same 4 points