Find the matrix $\mathbf{A}$ if $A\binom{7}{-1} = \binom{6}{2}.$ Find the $2\times2$ matrix $A$ where $A^2=A$ and
$$A\begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.$$
I tried plugging in: $A= \begin{pmatrix}a&b\\c&d\end{pmatrix}$ but that became messy very quickly. I got the equations:
$7a-b = 6$
$7c-d = 2$
$a^2+bc = a$
$ab+bd = b$
$ac + cd = c$
$bc + d^2 = d$ from trying that method. What should I do? 

Hint: Since you have $7a-b=6$ and $7c-d=2$ then $b=7a-6$ and $d=7c-2.$ Moreover, $A^2=A$ then $\det(A)^2=\det(A)$ which implies $\det(A)=0$ or $\det(A)=1$ then $ad=bc$ or $ad-bc=1.$ If $ad=bc$ then you get $a(7c-2)=c(7a-6)$ then  $-2a=-6c$ so  the coefficients of the matrix $A$ are of the form $a=3c,$ $b=7a-6=21c-6 $ and $d=7c-2.$ If $ad-bc=1$ then $-2a+6c=1$ so  the coefficients of the matrix $A$ are of the form $c=\frac{2a+1}{6}$ $b=7a-6$ and $d=7c-2=\frac{14a-5}{6}.$  
