Ali; starting with the equality $x = y$ for non-zero, equal real numbers x and y, follows the following steps in order. I. Let us multiply both sides of the equality by x: $$x ^ { 2 } = x \cdot y$$ II. Let us subtract $\mathrm { y } ^ { 2 }$ from both sides: $$x ^ { 2 } - y ^ { 2 } = x \cdot y - y ^ { 2 }$$ III. Let us factor both sides: $$( x + y ) ( x - y ) = y ( x - y )$$ IV. Let us divide both sides by $\mathrm { x } - \mathrm { y }$: $$x + y = y$$ V. Let us substitute y for x: $$2 y = y$$ As a result of these steps, Ali arrives at the conclusion "Every number equals twice itself." Accordingly, in which of the numbered steps did Ali make an error? A) I B) II C) III D) IV E) V
Ali; starting with the equality $x = y$ for non-zero, equal real numbers x and y, follows the following steps in order.
I. Let us multiply both sides of the equality by x:
$$x ^ { 2 } = x \cdot y$$
II. Let us subtract $\mathrm { y } ^ { 2 }$ from both sides:
$$x ^ { 2 } - y ^ { 2 } = x \cdot y - y ^ { 2 }$$
III. Let us factor both sides:
$$( x + y ) ( x - y ) = y ( x - y )$$
IV. Let us divide both sides by $\mathrm { x } - \mathrm { y }$:
$$x + y = y$$
V. Let us substitute y for x:
$$2 y = y$$
As a result of these steps, Ali arrives at the conclusion "Every number equals twice itself."
Accordingly, in which of the numbered steps did Ali make an error?
A) I
B) II
C) III
D) IV
E) V