How much is my partner worth?
x 2 + y 2 + x y x + y + x y x + y − x y = = = 1 8 6 ?
Don't be shy to post your solutions if you have a different method!
The answer is 3.
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5 solutions
my solution is this... x + y + √xy = 6 can be written as .. x + y = 6 - √xy............ Then by manipulating the first equation x² + y² + xy = 18... we can do.. x²+ y² + xy + xy = 18 + xy.. to have x² + 2xy + y² = 18 + xy or (x + y)² = 18 + xy........ by using (x + y)² = 18 + xy we can also use the value x+ y = 6 - √xy...... then ... (6-√xy)² = 18 + xy....... by expanding the value on the parenthesis.... 36 - 12√xy + xy = 18 + xy...... by combining like terms the remaining would be - 12√xy = -18...... there fore √xy = 3/2............. solving 2nd equation x + y + 3/2 = 6.. there fore x + y = 9/2......... solving the third equation x + y - √xy = ? ... given the value of x + y = 9/2 and √xy = 3/2.... the final equation is 9/2 - 3/2 = 6/2 or 3
Check your expansion of (x +y)^2. You should get: x^2+2xy+y^2.
For completeness, are there values of x and y that will satisfy your equations?
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x = 4 1 ( 9 + / − 3 5 ) and y = 4 1 ( 9 − / + 3 5 )
Very clever answer...Keep it up.
you just multiply the second and third equations then you'll find the result is 18/6 = 3
yeah right.. I'm thinking that the second to the last equation has no basis.... where did he get that one?
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It is just a random equation. You need two equations to determine the value of a third expression.
square the 2nd-equality , arrange to appear 1st-equality ->value of √xy -> sholve
x^2 + y^2+ xy = 18 (x + y)^2 = 18 + xy x + y = 6 - sqrt(xy) (6 - sqrt(xy))^2 = 18 + xy 36 - 12(sqrt(xy)) + xy = 18 + xy 36 - 12(sqrt(xy)) = 18 sqrt(xy) = 1.5 x + y = 4.5 x + y - sqrt(xy) = 3
x+y+sqrt(xy)=6 or sqrt(xy)-6=-(x+y), squaring both sides and rearranging yields x^2+xy+12sqrt(xy)+y^2=36. Subtract x^2+y^2+xy=18 from this yields 12sqrt(xy)=18 or sqrt(xy)=3/2 which yields x+y+3/2=6 or x+y=9/2. Finally x+y-sqrt(xy) = 9/2-3/2 = 3.
( x + y + x y ) ( x + y − x y ) = ( x + y ) 2 − x y = x 2 + y 2 + x y
6 ( x + y − x y ) = 1 8
x + y − x y = 6 1 8 = 3