Suppose there is a hyperplane in $\mathbb{R}^4$ that is the solution set to the homogeneous equation $x+2y-3z+w=0$ and a line in $\mathbb{R}^4$ given parametrically by $\{t(2,-3,0,0):t\in\mathbb{R}\}$.
Find a basis for the hyperplane and use the line to extend the basis for the hyperplane to a basis for $\mathbb{R}^4$
How could you find a basis for the hyperplane, and how would the line be useful in extending the basis for the hyperplane to a basis for $\mathbb{R}^4$?