Easy system of equations?

Question

Given the system of equations find $ v_3,v_4$ . If you only know the value of $ v_1,v_2$
$p_0+p=p_1$
$p_1+p=p_2$
$p_0+2p=p_3$
$p_3+p=p_4$
$p_1v_1 = p_2v_2 = p_3v_3 = p_4v_4$
Came to the equations when solving a complex physics task. Found that I don't know the method of solving such problems. I hope you will tell me one. Tried a lot of summing, subtracting equations one from each other, making substitutions, but did not succeded.
Thanks for help.

Answer 1

The first four linear equations have the solution $$ p=p_4-p_3,\; p_0=3p_3 - 2p_4,\; p_1=2p_3 - p_4,\; p_2=p_3 $$ with free parameters $p_3,p_4$. This follows just by adding and subtracting the equations. We can substitute this result to the system of polynomial equations $$ p_1v_1-p_2v_2=0, \; p_2v_2-p_3v_3=0, \; p_3v_3-p_4v_4=0. $$ Assume first that$p_3,p_4\neq 0$. Then it follows that $$ v_3=v_2,\; v_4= \frac{p_3v_2}{p_4},\; 2p_3v_1 - p_3v_2 - p_4v_1=0, $$ and we are done. For the subcases $p_3=0$ or $p_4=0$ it works similarly.