Graph systems of 2 or 3 linear equations and find intersection points visually. Shows parallel, coincident, and inconsistent cases.
Enter two or three linear equations and this tool draws each one, then marks every intersection with its coordinates and shows which of the three cases you are in: a single crossing, parallel lines that never meet, or coincident lines that are one and the same. Two-variable equations render as lines; three-variable ones as planes. The thing most users get wrong: no intersection is a valid answer, not an error. A system has zero, one, or infinitely many solutions - nothing else - and with three planes, every pair can intersect while no point lies on all three.
A concrete input and expected output from the current implementation.
Input
2x + y = 5 x - y = 1
Expected output
Intersection at (2, 1). One unique solution, consistent system: the lines have slopes -2 and 1, so they cross exactly once.
Substitute x = y + 1 from the second equation into the first: 2(y + 1) + y = 5, so 3y = 3, y = 1, and x = 2. Both equations check at (2, 1): 2(2) + 1 = 5 and 2 - 1 = 1. Because the slopes differ, the two lines cross in exactly one point.
Each equation becomes a geometric object: a straight line for two unknowns, a flat plane for three. Objects are drawn across a visible coordinate range, and wherever two of them cross, the crossing is marked with a point whose coordinates are shown. Nearly parallel lines cross at a glancing angle or far outside the default view, so the reported coordinates, not the on-screen picture, are what to trust.
A system of linear equations falls into exactly one of three cases, and the tool names the one you have: a unique solution when the coefficient determinant is nonzero; no solution when the lines are parallel or the planes never share a point; infinitely many solutions when the equations describe the same line or plane. The displayed label - consistent, inconsistent, or dependent - is the classification.
Lines that appear to cross on screen may be parallel in fact: equal slopes with different intercepts never meet, and plot resolution can make nearly parallel lines look as if they touch. Compare the slopes. If they are equal and the intercepts differ, no solution is the correct result, not an error.
The two equations describe the same line, so the graph shows a single line and every point on it satisfies both equations. Algebraically, one equation is a constant multiple of the other. Pick any x and both equations return the same y. Infinite solutions is a distinct outcome from no solution.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
Calculate percentages, percentage change, and what percent one number is of another with step-by-step breakdowns.
Calculate tips and split bills between multiple people with customizable tip percentages and rounding options.
Calculate monthly payments, total interest, and generate amortization schedules for loans and mortgages.
Calculate Body Mass Index from height and weight with category classification and healthy range guidance.
Calculate sale prices, savings amounts, and final costs after single or stacked percentage discounts.
Compute mean, median, mode, standard deviation, variance, and other descriptive statistics from a data set.