Systems of Linear Equations

A school district purchases a combination of 50 tablets and laptops for a new technology initiative. The total invoice arrives at $14,500. A tablet costs $200 and a laptop costs $450. Taken individually, these facts are merely constraints: one equation representing the total number of devices, and another representing the total budget. If we plot either constraint on a coordinate plane, we get an infinite continuum of possible scenarios. But when we impose both constraints on the universe at the same time, those infinite possibilities collapse into a single, unavoidable reality.

This is the essence of what mathematicians call a system of linear equations. Fundamentally, a system of linear equations consists of two or more linear equations containing the same variables. As a prospective middle school mathematics teacher, your task is to teach students how to navigate the transition between the physical reality of a problem, the algebraic manipulation of its symbols, and the geometric beauty of its graph.

Let us dissect the mechanics of these systems, how to solve them algebraically and graphically, and how these fundamental principles extend into the non-linear curves your students will encounter as they progress in their mathematical careers.