Online Equation Solver: Root Finder & Linear Systems
An equation solver for two tasks. Numerically find all real roots of a single equation f(x)=0 over a chosen interval, or solve a square linear system Ax=b from 2×2 up to 6×6 — with editable augmented-matrix cells, bulk paste of coefficients, and copy of the system. Results and validation appear instantly and everything runs locally in your browser.
Equation Solver Guide
Root finding and linear systems up to 6×6
The equation solver handles two problems: finding real roots of a single-variable equation f(x) = 0 over an interval, and solving a square linear system Ax = b of size 2×2 up to 6×6. Switch between the two modes, enter your equation or coefficients, and run the solver.
01 Single-equation mode
Enter an expression in x whose roots you want; the solver searches for values where f(x) = 0 within [xmin, xmax]. For example, entering x^2 − 4 over [−10, 10] finds both −2 and 2. The solver uses numerical bracketing, so roots must lie within the chosen interval; widen the range or split it if you expect more roots.
Linear-system mode solves Ax = b. Choose the matrix size (2×2 through 6×6), fill in the coefficients of A and the entries of b, and press Solve. The system is solved via row reduction; if it is singular or inconsistent, an error message explains the problem.
- Single mode supports the same math functions as the 2D plotter
- Root-finding works best when the interval bounds bracket a sign change
- Systems use Gaussian elimination (RREF)
- A singular or near-singular matrix has no unique solution
Mode tabs
- Single equationFind real roots of f(x) = 0 over an interval.
- Linear systemSolve a square system Ax = b.
Single-equation controls
- f(x) = 0Expression in x whose roots are sought.
- xmin / xmaxInterval to search for roots.
- Find rootsRuns numerical root-finding and lists solutions.
Linear-system controls
- size (n×n)Sets the system size from 2×2 to 6×6.
- matrix A cellsCoefficients of the left-hand matrix.
- vector b cellsRight-hand side constants.
- bulk text boxPaste or type the whole augmented matrix [A
- ApplyParses the bulk text box, sets the size and fills A and b in one step.
- CopyCopies the current augmented matrix as comma-separated rows.
- SolveSolves Ax = b and displays x₁ … xₙ.
- For single equations, plot f(x) first in the 2D grapher to see where roots lie.
- Widen the search interval if a known root is not found.
- Make sure every coefficient cell in a system is filled (use 0 for absent terms).
- For non-linear systems, solve equations one variable at a time.
A free online equation solver. Find all real roots of a single equation f(x) = 0 over an interval using numerical bisection, or solve a linear system Ax = b up to 6×6 via Gaussian elimination (RREF). Enter expressions with standard functions, see roots and solutions instantly. Runs locally in your browser.
Single-equation root finder
Enter an expression in x (it is solved as f(x)=0) and a search interval [a, b]. The solver scans the interval for sign changes and refines each root with bisection, listing every real root it finds. For example x^2-4 finds −2 and 2; sin(x) over [0, 6.28] finds 0, π and 2π.
Linear systems
Choose the number of equations (up to 6) and fill in the coefficient matrix A and the constants b. The system is solved by Gaussian elimination with partial pivoting, returning the reduced row echelon form and the solution vector x. Singular or inconsistent systems are reported.
Notes and limitations
The root finder catches sign-changing roots but cannot find roots where the function only touches zero without crossing (even multiplicity) unless the scan lands close by. Linear-system solving uses standard floating-point arithmetic; very ill-conditioned matrices may show small numerical residuals. All computation is local.
FAQ
QWhat equations can it solve?
Any single-variable expression f(x)=0 over a numeric interval, using standard functions (sin, cos, exp, log, sqrt, powers, etc.), plus linear systems Ax=b up to 6×6.
QHow are roots found?
The interval is scanned for sign changes and each crossing is refined with bisection, so multiple real roots in range are listed.
QWhy might a root be missing?
Roots where the function touches zero without crossing (even multiplicity) may not be detected by sign changes; widen or narrow the interval, or reformulate the equation.
QWhat size linear system is supported?
Up to 6 equations with 6 unknowns, solved by Gaussian elimination with partial pivoting (RREF).