Consider the line parametrized by \begin{align*} x&= 4t + 2,\\ y& = t+2. \end{align*}Find vector $\begin{pmatrix}a\\b\end{pmatrix}$ that's parallel to this line and satisfies $a+b = 10$.
I know that "a" is 4t+2 and "b" is t+2 but idk how to make it satisfy the equation a + b = 10.
$$(x,y)=(4t+2,t+2)=(4,1)t+(2,2)$$ That means vector $(a,b)$ is a multiple of $(4,1)$, since $(2,2)$ is just an offset. Then $$a=4k\\b=k\\a+b=10$$ Solve ang get $k=2$ and $$(a,b)=(8,2)$$