About This Dependent System Calculator
A dependent system appears when two linear equations describe the same line. That means every point on one equation also satisfies the other equation. This calculator checks that relationship with determinant tests, ratio checks, and clear result notes. It is useful for algebra practice, tutoring pages, worksheet builders, and quick classroom verification.
What the Tool Checks
The calculator compares two equations in standard form. The form is a1x + b1y = c1 and a2x + b2y = c2. It first finds the determinant of the coefficient matrix. A nonzero determinant means the lines cross once. That system is independent, not dependent. A zero determinant means the lines are either the same line or parallel lines. The tool then checks the augmented values. When all determinant values are zero, the equations are dependent and have infinitely many solutions. When the main determinant is zero but another determinant is not zero, the system is inconsistent and has no solution.
Why It Helps
Manual comparison can become confusing when negative signs, decimals, fractions, or zero coefficients appear. This page keeps the process organized. You can enter any real coefficients, choose a tolerance, set decimal precision, and rename variables. The response includes determinant values, proportional notes, classification, and a parametric solution when possible. The example table also shows common cases before users enter their own values. This makes the result easier to audit. It also helps learners see why a system has one solution, no solution, or infinitely many solutions.
Formula Used
For two equations, use D = a1b2 - a2b1. Also use Dx = c1b2 - c2b1 and Dy = a1c2 - a2c1. If D is not zero, x = Dx divided by D and y = Dy divided by D. If D, Dx, and Dy are all zero, the system is dependent. It has infinitely many ordered pair solutions.
How to Use This Calculator
Enter the coefficients from each equation. Keep each equation in standard form. Choose a tolerance for decimal comparisons. Select the number of decimal places for the output. Press calculate to view the result below the header. Use the CSV button to save a spreadsheet record. Use the PDF button to save a report for notes, lessons, or review. Teachers can reuse the exported values quickly.