One of the problems in one of the packets that I'm going through to review for a pre-test for an independent-study calculus class has asked me to derive the midpoint formula. I've gotten to the point where I have the following equation:
$$2 x_2 x_m - 2 x_m x_1 + 2 y_2 y_m - 2 y_m y_1 = x_2^2 - x_1^2 + y_2^2 - y_1^2$$
Would it be mathematically correct to split this into the following two equations…:
$$ \begin{cases} 2 x_2 x_m - 2 x_m x_1 = x_2^2 - x_1^2 \\ 2 y_2 y_2 - 2 y_m y_1 = y_2^2 - y_1^2 \end{cases} $$
…and treat them as a system of equations? If so, then how would I go about doing this?
You can not split this equation. That would be the same as saying $a+b=c+d$ can be split into $a=c$ and $b=d$.
If the points are $P_1, P_2, $ and $P_m$, for $P_m$ to be the midpoint between $P_1$ and $P_2$, then one way of writing this is that $P_m$ must be on the line between $P_1$ and$P_2$ (i.e., $P_m =tP_1+(1-t)P_2 $ for $0 \le t \le 1$) and $P_m$ must be at the halfway position (i.e., $t = \frac12$).
Another way is that $|P_1-P_m| = |P_2-P_m|$ and $|P_1-P_m| = |P_2-P_1|/2$ . This way gives you two equations for the two unknowns of the x and y components of $P_m$.