If x + 1 1 + y + 2 2 + z + 2 0 1 5 2 0 1 5 = 2 1 , what is x + 1 x + y + 2 y + z + 2 0 1 5 z ?
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I got it nice question
answer is 2.5 subtracting 3 from equatiom on both sides and solve the equation by taking -1 in each variables of the equation ,
THANK YOU! I AM REALLY CONVINCE BY YOUR SOLUTION.
Nice! So obvious once you see it!
x=1, y=-4, z=0
The first expression is 1 + 1 1 + − 4 + 2 2 + 0 + 2 0 1 5 2 0 1 5 = 2 1 − 1 + 1 = 2 1
The second is 1 + 1 1 + − 4 + 2 − 4 + 0 + 2 0 1 5 0 = 2 1 + − 2 + 0 = − 2 3
Any solution where z = x y − 2 x − y − 6 2 0 1 5 ( x y + 4 x + 3 y + 1 0 ) will work. There aren't just multiple solutions, there are uncountably infinite solutions.
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There may be infinite number of possible values for x, y, and z satisfying the first equation. But inserting the values of x, y and z (that satisfy the first equation) on the second equation will always yield to 2.5 as proven by Saunak. Your second expression has quite a mistake in basic division. The second fraction should be 2, not -2. Therefore, 0.5 + 2 + 0 = 2.5.
X=5 Z=2015*5 Y=10 also satifies!Nice solution!
We can simply satisfy 1/6 + 1/6 + 1/6 = 1/2. 1/(x+1) = 1/6, yields x = 5. 2/(y+2) = 1/6, yields y=10. 2015/(z+2015), yields z = 5(2015). The second sum becomes 5/(5+1) + 10/(10+2) + 5(2015)/(5(2015)+2015) = 5/6 + 5/6 + 5/6 = 15/6 = 2.5.
Simplify the equation x+1/x+1 + y+2/y+2 + z+2015/z+2015 - (1/x+1 + 2/y+2 + 2015/z+2015 ) = 3 - 1/2 = 2.5
I solved it by inspection...z must equal infinity to zero out its fraction...putting x equal to 0 yields 1 and putting y equal to -6 yield -1/2 so it comes out to 1/2. Subbing those values in the second equation yields 2.5
Why do you assume that x must be zero - it is a lucky guess in this case, but the logic of how you got there escapes me - especially given your first statement that x must be infinity.
Well, if y=z=0 then x=-5/3, so x/(x+1)=5/2.
Presumably this works for any values of x , y , and z that make the first equation true, so any convenient values can be chosen for x , y , and z , such as x = 1 , y = − 4 , and z = 0 , and the desired expression computed as 2 1 + 2 + 0 .
Putting y=0,z=0 into the first equation gives x=-5/3. Then putting y=0,z=0, x=-5/3 into the second gives the result
1-(1/(1+x)+1-2/(y+2)+1-z/(z+2015)=3-(1/1+x)+2/(y+2)+z/(z+2015))=3-(1/2)=2.5
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Adding the two fractional expressions gives x + 1 x + 1 + y + 2 y + 2 + z + 2 0 1 5 z + 2 0 1 5 = 3 , so our unknown value must be 3 − 2 1 = 2 5 = 2 . 5 .