In which step is the error?

Step 1: Let x = 1 and y = 1.
Step 2: By substitution xy = y^2.
Step 3: Multiplying both sides by -1, -xy= -y^2.
Step 4: Adding x^2 to both sides, x^2-xy=x^2-y^2.
Step 5: Factoring, x(x-y)=(x+y)(x-y).
Step 6: Dividing both sides by x-y, x=x+y.
Step 7. By substitution, 1=1+1
Step 8: 1 = 2.

step 6

x = y = 1

so division by x-y is division by zero
a BIG no-no

The error is in Step 6. Let's analyze the steps leading up to it and explain the mistake.

In Step 1, the starting values for x and y are given as x = 1 and y = 1.

In Step 2, substitution is used to replace xy with y^2. This step is correct.

In Step 3, both sides of the equation are multiplied by -1. The purpose of this step is to change the signs of the terms. The equation becomes -xy = -y^2, and this step is correct.

In Step 4, x^2 is added to both sides of the equation. The equation becomes x^2 - xy = x^2 - y^2. This step is correct.

In Step 5, factoring is applied to both sides of the equation. The left side is factored as x(x - y), and the right side is factored as (x + y)(x - y). This step is correct.

Now, in Step 6, a mistake occurs. The division is done by (x - y) on both sides of the equation. The equation becomes x = x + y. However, dividing by (x - y) is not allowed since x - y = 0, which would result in division by zero. This is where the error lies.

Due to the mistake in Step 6, subsequent steps are affected.

In Step 7, substitution is done to replace x with 1 and y with 1. The equation becomes 1 = 1 + 1, which simplifies to 1 = 2.

In Step 8, the incorrect equation 1 = 2 is presented. This result is a consequence of the error made in Step 6.

To correct the error, Step 6 should be revisited and a different approach should be used to simplify the equation.