Given that $\vec{a}=\langle 10,4\rangle$ and $\vec{n}=\langle 3,9\rangle$ find $-4 \vec{a}+5 \vec{n}$

Step 1 ：We are given two vectors, \(\vec{a} = \langle 10,4 \rangle\) and \(\vec{n} = \langle 3,9 \rangle\).

Step 2 ：We are asked to find the result of the operation \(-4 \vec{a} + 5 \vec{n}\).

Step 3 ：To do this, we first multiply each vector by its corresponding scalar. This gives us \(-4 \vec{a} = \langle -40,-16 \rangle\) and \(5 \vec{n} = \langle 15,45 \rangle\).

Step 4 ：We then add these two vectors together to get the result. This gives us \(\langle -40,-16 \rangle + \langle 15,45 \rangle = \langle -25,29 \rangle\).

Step 5 ：So, the result of the operation \(-4 \vec{a} + 5 \vec{n}\) is the vector \(\boxed{\langle -25,29 \rangle}\).