Modelling Two Constraints with a System

Two constraints on the same pair of quantities means two equations in the same two variables. Each sentence in the problem is a separate limit, one on how many, one on how much, and each one becomes its own equation. The trick is naming: pick one letter for one quantity, and if the problem gives you a total, let the second quantity be that total minus your first letter. That way both equations end up written in the same two letters instead of drifting into different names that cannot be solved together.

Once both constraints are written in the same letters, you have a system, and a system with two different linear equations in two unknowns has exactly one solution, the single pair of values that satisfies both limits at once. That solution is the one workable plan: the count of each type of interval, the number of each item bought, whatever the two quantities represent. You solve it the normal way, by substitution if one variable is already isolated, or by combining the equations to cancel one letter.

This sits right next to systems of linear equations in general, but the modelling step, turning words into matching variables, is the real skill being tested here. It also connects forward to what happens when a third constraint shows up: two equations already fix both unknowns, so a third equation is an extra demand on a plan that is already decided. Unless that third line happens to pass through the same point, the system becomes inconsistent and has no solution at all.