Consider the general first-order initial value problem y'(t) = ay + b, y(0) = y₀, for t ≥ 0, where a, b, and y₀ are real numbers.

a. Explain why y = -b/a is an equilibrium solution and corresponds to horizontal line segments in the direction field.

b. Draw a representative direction field in the case that a > 0. Show that if y₀ > -b/a, then the solution increases for t ≥ 0 and if y₀ < -b/a, then the solution decreases for t ≥ 0.

c. Draw a representative direction field in the case that a < 0. Show that if y₀ > -b/a, then the solution decreases for t ≥ 0 and if y₀ < -b/a, then the solution increases for t ≥ 0.