Proof that 2=1

Algebra Level 2

The following are my steps when I proved that 2 = 1 2=1 .

Step 1. Suppose that : : y = b y=b

Step 2. Multiply both sides by y y : : y 2 = y b y^2=yb

Step 3. Subtract b 2 b^2 from both sides : : y 2 b 2 = y b b 2 y^2-b^2=yb-b^2

Step 4. Factor both sides : : ( y b ) ( y + b ) = b ( y b ) (y-b)(y+b)=b(y-b)

Step 5. Divide both sides by y b y-b : : y + b = b y+b=b

Step 6. Since y = b y=b , substitute : : b + b = b b+b=b or 2 b = b 2b=b

Step 7. Divide both sides by b b : : 2 = 1 2=1

Is there something with my calculation? If your answer is n o no , give your answer as 10 10 . If your answer is y e s yes , input the step number which I first committed a mistake.

6 2 10 4 3 7 1 5

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2 solutions

Noel Lo
Jul 17, 2018

Every step is valid up till step 5. Considering that y = b y=b , y b = 0 y-b=0 and dividing by zero is invalid.

In step 5 5 , we divided both sides of the equation by 0 0 , because y b = 0 y-b=0 if y = b y=b . Hence, step 5 5 , 6 6 and 7 7 are not valid, because division by zero is not allowed. Therefore, the answer is 5 5 , since I committed the first mistake on step 5 5 .

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