SOLUTION: The following appears to prove that any two numbers are equal. Obviously it is wrong; can you spot the flaw?
Let a and b be any two different numbers. Define x as the differenc
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-> SOLUTION: The following appears to prove that any two numbers are equal. Obviously it is wrong; can you spot the flaw?
Let a and b be any two different numbers. Define x as the differenc
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Question 329612: The following appears to prove that any two numbers are equal. Obviously it is wrong; can you spot the flaw?
Let a and b be any two different numbers. Define x as the difference between them:
x = b - a
Multiply both sides of the equation by (b-a):
x(b – a) = (b – a)(b – a)
bx – ax = b2 – 2ab + a2
Add –bx+ab-a2 to both sides and simplify:
bx – ax – bx + ab – a2 = b2 – 2ab + a2 – bx + ab – a2
Both sides of this equation have a common factor:
a(–x + b – a) = b(–x + b – a)
Divide both sides by (-x+b-a):
a(–x + b – a) = b(–x + b – a)
–x + b – a –x + b – a
a = b
What was my mistake? Found 3 solutions by stanbon, galactus, Theo:Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! Since x = b-a
-x = a-b
-x+b-a = 0
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When you divided by -x+b-a
you divided by zero; that
fraction is undefined.
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Cheers,
Stan H.
You can put this solution on YOUR website! Your error is in dividing by 0.
If x=a-b, then -x+a-b ---> -(a-b)+(a-b)=0
There a many of these types of mathematical fallacies. One of the 'errors' they have in common is division by 0.
–ax + ab – a2 = –bx + b2 – ab
Both sides of this equation have a common factor:
a(–x + b – a) = b(–x + b – a)
Divide both sides by (-x+b-a):
a(–x + b – a) = b(–x + b – a)
–x + b – a –x + b – a
a = b
You got the right solution because you did all the intermediate steps correctly.
The flaw is as follows:
Start from:
a(–x + b – a) = b(–x + b – a)
Substitute (b-a) for x since this is the premise that you started with.
You get:
a * (-(b-a) + b - a) = b * (-(b-a) + b - a)
Simplify to get:
a * (-b + a + b - a) = b * (-b + a + b - a)
simplify further to get:
a * 0 = b * 0
Which will always be true regardless of the values of a or b.
Furthermore, you cannot divide by (-b + a + b - a) because division by 0 is not allowed.
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