Graphing Differential Vector Equations

Question

My linear algebra book is discussing differential vector equations, and they give the example function $x(t) = (-3/70)\begin{bmatrix}-5\\2\end{bmatrix}e^{6t} + (188/70)\begin{bmatrix}1\\1\end{bmatrix}e^{-t}$. They then graph the function as follows: The thing is, when I plug numbers in for $t$, and graph by hand, I only get one line -- not all these lines as they show on this graph and similar graphs. So I'm confused on what they're doing here. Can somebody explain to me how to properly graph these equations? Thank you.