Find x and y

[nycmathguy]

Find x and y.

 

[MathLover1]

here we have to much work to do, so I will post short version

x+y-sqrt(xy)=7.........eq.1

x^2+y^2+xy=133..........eq.2
-----------------------------------

x^2+y^2+xy=133..........eq.2, isolate y
y^2+xy=133-x^2 ........complete square on left side
(y^2+xy+b^2)-b^2=133-x^2.........b=x/2
(y^2+xy+(x/2)^2)-(x/2)^2=133-x^2

(y+x/2)^2-x^2/4=133-x^2
(y+x/2)^2=133-x^2+x^2/4
(y+x/2)^2=133-(3/4)x^2
(y+x/2)^2=1/4 (532 - 3 x^2)
y+x/2=sqrt(1/4(532 - 3x^2))

y=sqrt(532 - 3x^2)/2-x/2

y=(sqrt(532 - 3x^2)-x)/2............eq.2a

substitute in eq.1

x+(sqrt(532 - 3x^2)-x)/2-sqrt(x((sqrt(532 - 3x^2)-x)/2))=7.........eq.1

ones you solve it for x, you get solutions

x = 4
x = 9

x^2+y^2+xy=133..........eq.2, substitute x=4
4^2+y^2+4y=133
16+y^2+4y=133
y^2+4y=133 -16
y^2+4y=117
y^2+4y-117=0
(y - 9) (y + 13) = 0

y=9
y=-13

if x=9
9^2+y^2+9y=133
y=4
y = -13

solutions:

x=4, y=9
x=9, y=4

 

[nycmathguy]

here we have to much work to do, so I will post short version

x+y-sqrt(xy)=7.........eq.1

x^2+y^2+xy=133..........eq.2
-----------------------------------

x^2+y^2+xy=133..........eq.2, isolate y
y^2+xy=133-x^2 ........complete square on left side
(y^2+xy+b^2)-b^2=133-x^2.........b=x/2
(y^2+xy+(x/2)^2)-(x/2)^2=133-x^2

(y+x/2)^2-x^2/4=133-x^2
(y+x/2)^2=133-x^2+x^2/4
(y+x/2)^2=133-(3/4)x^2
(y+x/2)^2=1/4 (532 - 3 x^2)
y+x/2=sqrt(1/4(532 - 3x^2))

y=sqrt(532 - 3x^2)/2-x/2

y=(sqrt(532 - 3x^2)-x)/2............eq.2a

substitute in eq.1

x+(sqrt(532 - 3x^2)-x)/2-sqrt(x((sqrt(532 - 3x^2)-x)/2))=7.........eq.1

ones you solve it for x, you get solutions

x = 4
x = 9

x^2+y^2+xy=133..........eq.2, substitute x=4
4^2+y^2+4y=133
16+y^2+4y=133
y^2+4y=133 -16
y^2+4y=117
y^2+4y-117=0
(y - 9) (y + 13) = 0

y=9
y=-13

if x=9
9^2+y^2+9y=133
y=4
y = -13

solutions:

x=4, y=9
x=9, y=4

Wow! Tough one here.

 

[MathLover1]

yes, and please do not look for problems of this kind
solving them is time consuming and does not help you much

 

[nycmathguy]

yes, and please do not look for problems of this kind
solving them is time consuming and does not help you much

Ok. Will do.