Let $G$ be the group of invertible upper triangular $2 \times 2$ matrices with real entries and let $H$ be the subset of $G$ consisting of those matrices $A$ with $a_{11} = 1$. It is easy to see that $H$ is a normal subgroup of $G$.
What is eluding me is how to construct an epimorphism for which $H$ is the kernel so that I can identify the structure of $G/H$ using the First Isomorphism Theorem.
Consider the map $\phi: G \to \mathbb{R}^{\star}$ (where $\mathbb{R}^{\star}$ is the group of real numbers without $0$ under multiplication): $$ \begin{pmatrix} a & b \\ 0& c\end{pmatrix} \mapsto a$$ You can check that this a surjective (since $a \in \mathbb{R}\setminus\{0\}$) homomorphism with kernel $H$, hence $G/H \cong \mathbb{R}^{\star}$