Let's prove that $1 =2$ by using two variables, $a$ and $b$ . Before we start, let's let $a=b$ . Now, we can prove that $1=2$ . $1) \ a = b \ || \ \text{Given}$

$2) \ a^2=ab \ || \ \text{Multiply both sides by} \ a$

$3) \ a^2-b^2=ab-b^2 \ || \ \text{Subtract both sides by} \ b^2$

$4) \ (a+b)(a-b)=b(a-b) \ || \ \text{Factor both sides}$

$5) \ (a+b)=b \ || \ \text{Divide both sides by} \ (a-b)$

$6) \ a + a = a \ || \ \text{Substitute} \ a \ \text{for} \ b$

$7) \ 2a = a \ || \ \text{Addition}$

$8) \ 2 = 1 \ || \ \text{Divide both sides by} \ a$

Which step is incorrect?

Step
$4$
Step
$3$
Step
$5$
The proof is correct.

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At step 5, we are dividing by (a-b) but as stated in 1, a=b so we are dividing by 0. This doesn't make sense and is considered undefined. On the previous steps no divisions had been made and they are logical progressions so no earlier mistakes are made.