Modeling Linear Equations & Inequalities — Module Hub & Visuals
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.
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Lesson Slides & PDFs
Student-facing lesson slide PDFs are grouped by topic below for review, catch-up, and family support.
📖 Student Edition for this module (PDF) ✍ Skills Practice — Student Edition (PDF)
Graphic Organizers & Activities
Foldable notes and activities for each lesson, with the one to start with listed first. Sign in with your alias and PIN to open them.
Start Here — What Module 3 Is About
What this module is for, and what you will be able to do by the end of it.
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.
Key Vocabulary
The precise words for what you just investigated. The standards this module carries are in the next section.
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.
Standards in This Module
The Texas Essential Knowledge and Skills this module is built to teach. RDY marks a STAAR readiness standard.
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.
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.
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.
Want to See This in 3D?
Optional, and always second. The lab above is where you measure and calculate; this is the same idea as an everyday object you can turn over and inspect. Open it when the lab already makes sense — or skip it entirely.
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.