1. **Problem statement:** We are given two lines: - Line $g$ in parametric form: $$\mathbf{X} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + t \cdot \begin{pmatrix} 1 \\ 4 \end{pmatrix}$$ - Line $h$ in slope-intercept form: $$y = kx + 7$$ We need to find the value of $k$ such that lines $g$ and $h$ are parallel. 2. **Formula and rules:** - Two lines are parallel if their direction vectors have the same slope. - The slope of line $h$ is directly given by $k$. - The direction vector of line $g$ is $$\begin{pmatrix} 1 \\ 4 \end{pmatrix}$$, so its slope is $$\frac{\text{change in } y}{\text{change in } x} = \frac{4}{1} = 4$$. 3. **Find $k$:** Since $g$ and $h$ are parallel, their slopes must be equal: $$k = 4$$ 4. **Answer:** The value of $k$ that makes $g$ and $h$ parallel is $$\boxed{4}$$. This means the line $h$ has the equation $$y = 4x + 7$$ and is parallel to the line $g$.