Final answer:
To choose h and k such that the system has no solution, a unique solution, and many solutions, consider the number of solutions of a system of linear equations in two variables.
Step-by-step explanation:
In order to choose h and k such that the system has no solution, a unique solution, and many solutions, we need to consider the number of solutions of a system of linear equations in two variables.
If the system has no solution, it means the lines representing the equations are parallel and will never intersect. This can be achieved when their slopes are equal but their y-intercepts are different, for example, h = 2 and k = 3.
If the system has a unique solution, it means the lines representing the equations intersect at a single point. This can be achieved when the slopes and y-intercepts of the lines are different, for example, h = 2 and k = 5.
If the system has many solutions, it means the lines representing the equations are identical and overlap. This can be achieved when the lines have the same slope and y-intercept, for example, h = 2 and k = 2.