STEM REASONING AND EVIDENCE WORKBENCH
Classify two linear equations as unique, parallel, or coincident and audit learner intersection evidence with determinants.
Review output
Load the worked example to inspect realistic inputs, or enter anonymous classroom or laboratory evidence. Results remain in this browser tab until you copy or download them.
Transparent method
For a1x plus b1y equals c1 and a2x plus b2y equals c2, determinant D equals a1b2 minus a2b1. If D is nonzero, x equals the c-column determinant divided by D and y equals the y-column determinant divided by D. If D is zero, matching augmented determinants mean coincident lines; otherwise they are parallel. Supported means both the system classification and any required unique coordinates agree. Parallel lines have no common point, coincident equations describe the same entire line, and a unique system has one intersection that must satisfy both originals. Use the determinant as evidence, then connect it with elimination steps and a graph.
Confirm the question, approved source, units, notation, signs, response coding, model assumptions, and review threshold. Keep an unchanged copy of measurements or learner evidence.
Run the worked example, inspect every intermediate row, and compare the result with a hand calculation, diagram, representation, periodic table, or approved laboratory method.
Record the method, assumptions, uncertainty, units, classifications, and flagged rows. Verify the editable report independently before instruction, assessment, or laboratory use.
How to interpret the result
Use after substitution or elimination work when learners may confuse no solution with infinitely many solutions, stop after dependent equations, or report an unverified point. The engine supplies a coefficient-level classification and exact numerical intersection. It does not parse free-form equations, solve nonlinear systems, handle inequalities, determine contextual feasibility, or replace shown reasoning.
The first system intersects at three comma two. The second has proportional variable coefficients but a conflicting constant, so its lines are parallel. The third equation is exactly three times the first and is coincident. The final system actually intersects at three comma two, so the learner coordinates two comma three fail substitution.
For a1x plus b1y equals c1 and a2x plus b2y equals c2, determinant D equals a1b2 minus a2b1. If D is nonzero, x equals the c-column determinant divided by D and y equals the y-column determinant divided by D. If D is zero, matching augmented determinants mean coincident lines; otherwise they are parallel.
Supported means both the system classification and any required unique coordinates agree. Parallel lines have no common point, coincident equations describe the same entire line, and a unique system has one intersection that must satisfy both originals. Use the determinant as evidence, then connect it with elimination steps and a graph.
Do not divide by a zero determinant, call every canceled-variable result no solution, invent one point for coincident equations, swap x and y, lose signs in elimination, or verify a point in only one equation. Preserve equation balance, interpret the remaining statement, graph when useful, and substitute unique coordinates into both equations.
Learning boundary
This browser-local workbench applies only the disclosed rules. It cannot diagnose a learner, certify an experiment, establish a chemical structure, or replace teacher, laboratory, and safety judgment.
Review checklist
Practical questions
No. Processing occurs in the current browser tab. Avoid personal information and retain authoritative records only in the school-approved system.
No. It describes entered evidence under one disclosed rule. Compare several evidence sources and ask for reasoning before drawing a conclusion.
Treat it as an editable calculation aid. Show your own reasoning, cite required sources, and follow the course method and academic-integrity rules.
YOUR NIRMION
