True or False? 1) Let x and y real numbers. If x^2-5x=y^2-5y and x != y, then x+y is five. 2) The real number pi can be expressed as a repeating decimal. 3) If an irrational number is divided by a nonzero integer the result is irrational.

Wierzycaz 2020-12-24 Answered
True or False?
1) Let x and y real numbers. If x25x=y25y and xy, then x+y is five.
2) The real number pi can be expressed as a repeating decimal.
3) If an irrational number is divided by a nonzero integer the result is irrational.
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Expert Answer

hajavaF
Answered 2020-12-25 Author has 90 answers
1) True.
x25x=y25y
Rearranging equation,
x2y2=5x5y
x2y2=5(xy)
Using identity x2y2=(x+y)(xy) we get,
(x+y)(xy)=5(xy)
Dividing by (x−y) on both sides we get,
x+y=5
Therefore, if x25x=y25y and xy, then x+y is five.
2) False.
The decimal expansion of pi is 3.1415926535897932384....., the decimal does not repeat exactly. pi is an irrational number. Therefore, it cannot be expressed as ratio of two integers and it cannot be expressed as terminating or repeating decimal.
3) True.
Any irrational number divided by an integer is irrational number. Proof (by contradiction):
Let x be a irrational number and a be an integer if xa is a rational number it can be expressed as ratio of two integers, xa=mn this implies x=amn. Since a, m and n are integers, am is an integer and x is ratio of two integers which implies x is a rational number. This is a contradiction to fact that x is irrational number.
Therefore, if an irrational number divided by a non zero integer then the result is an irrational number.
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