Solve Gm divided by d squared equals Gx divided by d squared, for x.

Step-by-step solution

1. 1

   Record the conditions that make the algebra legal. The equation only makes sense if $d\neq 0$ (otherwise both sides are undefined), and dividing by $G$ later requires $G\neq 0$. In the physical reading — this is Newton's gravitational field $g=Gm/d^2$ compared at the same distance — $G=6.674\times 10^{-11}$ is a nonzero constant, so both conditions hold automatically.

2. 2

   Clear the identical denominators. Multiplying both sides by $d^2$:

   $Gm=Gx$

   Because the same $d^2$ sits under both sides, one multiplication removes it entirely; there is no cross-multiplication to do.

3. 3

   Divide by the common factor $G$.

   $m=x$

   This is the only step where $G\neq 0$ is needed. Note that no rearranging of $m$ or $x$ was required — they were already isolated by the cancellations.

4. 4

   Interpret the result. The distance $d$ has dropped out completely, so the conclusion holds at every separation: two bodies produce the same gravitational field at the same distance precisely when their masses are equal. The answer therefore depends on neither $d$ nor $G$.

5. 5

   Check by substitution. Putting $x=m$ back in makes the two sides literally the same expression $\dfrac{Gm}{d^2}$, so the equation is satisfied — and it is satisfied for no other value of $x$, since $Gm=Gx$ forces $x=m$ once $G\neq 0$.