System of equations

Given $\begin{cases} 2a + b + c + d + e = 6 & ...(1) \\ a + 2b + c + d + e = 12 & ...(2) \\ a + b + 2c + d + e = 24 & ...(3) \\ a + b + c + 2d + e = 48 & ...(4) \\ a + b + c + d + 2e = 96 & ...(5) \end{cases}$

$\begin{array} {lrll} (1)+(2)+(3)+(4)+(5): & 6a+6b+6c+6d+6e & =186 \\ & \implies a+b+c+d+e & = 31 &...(6) \\ (1)-(2)+(3)-(4)+(5): & 2a+2c+2e & =66 &...(7) \\ (7)-(6): & \implies a-b+c-d+e & = \boxed{35} \end{array}$

Adding the five equations we get,

$6a+6b+6c+6d+6e=186$

$a+b+c+d+e=31$ $\implies$ $\boxed{6}$

$\boxed{1}-\boxed{6}$

$a=6-31=-25$

$\boxed{2}-\boxed{6}$

$b=12-31=-19$

$\boxed{3}-\boxed{6}$

$c=24-31=-7$

$\boxed{4}-\boxed{6}$

$d=48-31=17$

$\boxed{5}-\boxed{6}$

$e=96-31=65$

Finally,

$a-b+c-d+e=-25-(-19)+(-7)-17+65=$ $\boxed{35}$