Answer
$\mathbf{v}=-\mathbf{u}+2\mathbf{w}$
Work Step by Step
Write the vector $\mathbf{v}=(1,-4)$ as a linear combination of $\mathbf{u}=(1,2)$ and $\mathbf{w}=(1,-1)$
If
$\mathbf{v}=a\mathbf{u}+b\mathbf{w}$
Then
$1=a+b$
$-4=2a-b$
If we add these equations together we get
$-3=3a\implies a=-1$
Substituting this value back into our first equation gives us
$1=-1+b\implies b=2$
This is consistent with our second equation:
$-4=2\times(-1)-2$
So our linear combination is
$\mathbf{v}=-\mathbf{u}+2\mathbf{w}$