Modeling Linear Equations & Inequalities — Visual Lab
Two lines, two sliders, one big question: where do they agree? Move the parts and watch the solution appear — a single point, no point, an entire line, or a whole region of the plane.
In Module 3 a solution stops being a single number and becomes a place. For a system of equations it is the point where two lines cross. For a system of inequalities it grows into a shaded feasible region — every \((x,\,y)\) that satisfies all the constraints at once. This lab lets you build both with your hands.
Systems & Inequalities Visualizer
Drag the sliders to reshape each line. Flip to Inequalities to shade the half-planes and reveal the region where both are true.
Two ways to read "solution." In Equations mode the marked point is the one ordered pair that works in both equations. In Inequalities mode the deep-green overlap is the feasible region — the set of every point that satisfies both constraints.
What You're Seeing · Try This
Read the left panel first, then run the experiments on the right. Predict before you drag.
What you're seeing
Each line is written \(y = mx + b\): the slider \(m\) tilts it (rate of change) and \(b\) slides it up or down (the \(y\)-intercept).
Equations mode draws both lines and marks where they cross. The readout names the solution type: one point (lines with different slopes), none (parallel lines), or infinitely many (the same line).
Inequalities mode turns each line into a boundary. A dashed edge means strict (\(<,\ >\)) — not included; a solid edge means \(\le\) or \(\ge\) — included. The overlap of the two shaded half-planes is the feasible region.
Try this
- Set both slopes different (say \(m=2\) and \(m=-1\)). Find the single crossing point — that ordered pair is the one solution.
- Make the slopes equal but the intercepts different. The lines go parallel — the readout flips to No solution. Why can't parallel lines meet?
- Now match the intercepts too, so the lines coincide. Watch it become infinitely many solutions — one line hiding two equations.
- Switch to Inequalities. Set Line 1 to \(y \le\) and Line 2 to \(y \ge\), then hunt for the green feasible region. Change one to a strict \(<\) and notice the boundary turn dashed.
Key Vocabulary & Standards
The words a mathematician uses for what you just built — and the TEKS this lab puts into action.
Two or more equations (or inequalities) considered together. A solution must work in all of them at once.
The ordered pair (or set of pairs) that makes every equation true — geometrically, the intersection.
A system with at least one solution is consistent; parallel lines that never meet are inconsistent.
One crossing point is independent; two equations describing the same line are dependent (infinitely many solutions).
The region of the plane on one side of a boundary line — the graph of a single linear inequality.
The overlap of all half-planes in a system of inequalities: every point that satisfies all constraints at once.
Boundary classification (\(<,\ \le,\ >,\ \ge\)) and feasible-region reasoning align to the TEA Bluebonnet Learning — Secondary Mathematics open curriculum, licensed CC BY-NC 4.0. Non-commercial classroom use.
Where to Go Next
Take the idea back to the course documents, or keep exploring.
Instructor: Dr. Goodluck Ijezie-Desbois, PharmD · Beta Academy · Room: TBA
Reach out by appointment, at gijezie-desbois@betaacademy.org, or through ParentSquare.