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 \(\boxed{-49\mathbf{i} - 7\mathbf{j}}\).