Let $\mathbf{v}=-8 \mathbf{i}+4 \mathbf{j}$ and $\mathbf{w}=-\mathbf{i}-3 \mathbf{j}$. Find $5 \mathbf{v}-9 \mathbf{w}$.
$5 v-9 w=\square$ (Simplify your answer. Type your answer in terms of $i$ and j.)

Step 1 ：Let's start by multiplying each vector by their respective scalar. For vector v, we multiply it by 5, and for vector w, we multiply it by -9.

Step 2 ：Doing this, we get: \(5\mathbf{v} = 5(-8\mathbf{i}+4\mathbf{j}) = -40\mathbf{i}+20\mathbf{j}\) and \(-9\mathbf{w} = -9(-\mathbf{i}-3\mathbf{j}) = 9\mathbf{i}+27\mathbf{j}\).

Step 3 ：Now, we subtract the two results to get the final vector: \(5\mathbf{v}-9\mathbf{w} = (-40\mathbf{i}+20\mathbf{j}) - (9\mathbf{i}+27\mathbf{j})\).

Step 4 ：Simplifying this, we get: \(-40\mathbf{i}+20\mathbf{j} - 9\mathbf{i}-27\mathbf{j} = -49\mathbf{i} - 7\mathbf{j}\).

Step 5 ：Final Answer: The result of the operation \(5\mathbf{v}-9\mathbf{w}\) is \(\boxed{-49\mathbf{i} - 7\mathbf{j}}\).