How do you write a system of equations with the solution (4,-3)?

You have given a point coordinate. There are infinite equations that will have that solution. Remember, you need at least two points to define a line.

The most obvious set of equations with ##(4,-3)## as a solution would be ##x=4## ##y=-3##

If you want to restrict yourself to linear equations, you could use the slope-point form: ##y-(-3) = m(x-4)## for two (different) arbitrary values for ##m##, the slope. for example ##m_1 = 2## and ##m_2 = 3## would give equations ##y+3 = 2x-8## or equivalently ##y=2x-11## and ##y+3 = 3x-12## or equivalently ##y=3x-15## which together would have the required ##(4,-3)## solution.