Ten is Twenty

Algebra Level 1

Here is a proof that 10 = 20 : 10 = 20:

Step 1: Consider the system of equations 2 x + 4 y = 20 x + 2 y = 5. 2x + 4y = 20 \\ x + 2 y = 5. Step 2: We can rewrite the second equation as x = 5 − 2 y . x = 5 - 2y. Step 3: Now, plugging x = 5 − 2 y x=5 - 2y into the first equation, we get 2 ( 5 − 2 y ) + 4 y = 20. 2(5-2y) + 4y = 20. Step 4: Simplifying, we have 10 − 4 y + 4 y = 20. 10 - 4y + 4y = 20. Step 5: Cancelling out the two 4 y 4y 's, we get 10 = 20. 10 = 20.

Where did the proof go wrong?

The two 4 y 4y 's can't be cancelled out in step 4 The system of equations has no solutions in step 1 There is an arithmetic error somewhere in the 5 steps The reasoning is correct all the way through the 5 steps

Agnishom Chattopadhyay by Agnishom Chattopadhyay , 22, India

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2 solutions

Saksham Jain
Nov 19, 2017

2 x + 4 y = 20 x + 2 y = 5 \huge 2x + 4y = 20 \\\huge x + 2 y = 5 as we can see if we multiply 2nd equation by 2.we get these equations \huge \text{as we can see if we multiply 2nd equation by 2.we get these equations} 2 x + 4 y = 20 2 x + 4 y = 10 \huge 2x + 4y = 20 \\\huge 2x + 4 y = 10 therefore mistake was made in 1 st step as 10=20 was already taken \huge \text{therefore mistake was made in 1 st step as 10=20 was already taken}

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Krishna Karthik
Mar 16, 2019

Equation 2 just contradicts equation 1. Equation two says 2x+4y=10, whereas equation 1 says 2x+4y=20.

Also, since the two lines of the equations are parallel (slopes are the same), then they don't have any intersection points, meaning that they don't have any solutions.

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