Simultaneous Equation Solver
Solve systems of two or three linear equations with the elimination method shown step by step. Fractional solutions appear in exact reduced form, and inconsistent or dependent systems are detected. Ideal for checking school algebra homework.
Try an example
Enter the coefficients of each equation to see the solution with the elimination working. Fractional solutions are shown exactly.
How to use
- 1
Choose the size
Pick 2 variables (x, y) or 3 (x, y, z) and enter each equation's coefficients.
- 2
Read the solution
The elimination and back-substitution working is shown, with exact fractional results.
- 3
Check special cases
Inconsistent (no solution) and dependent (infinitely many) systems are reported with the reason.
Features
- Systems of 2 or 3 linear equations solved with elimination working
- Solutions shown and copyable as exact reduced fractions with decimals
- Inconsistent and dependent systems detected automatically with the reason
- Inputs are never transmitted; all computation happens locally in your browser
Use cases
Check your homework
Work the elimination by hand and confirm the solution only.
Intersections of lines
Turn two lines y = ax + b into a system and find the crossing point.
Practice 3-variable systems
Follow the elimination steps of high-school 3-variable systems.
Details
A system of linear equations asks for variable values that satisfy every equation at once. For two variables it is solved by elimination (multiplying and adding/subtracting to remove one variable) or substitution; this tool shows the elimination working.
All arithmetic is exact over fractions. For example, 3x + 2y = 12 with x − y = 1 gives x = 14/5 and y = 9/5 with no rounding error. Decimal coefficients such as 0.5 are converted to exact fractions before computing.
If the coefficient ratios match but the constant ratios differ, the system has no solution (parallel lines in 2D); if every ratio matches, there are infinitely many solutions (the same line or plane). Keep coefficient absolute values at or below 1,000,000. Inputs are never transmitted; all computation happens locally in your browser.
FAQ
Does it use elimination or substitution?
Elimination. Both equations are multiplied by suitable numbers and subtracted to remove one variable; the remaining one-variable equation is solved and substituted back into an original equation. The tool shows this whole sequence.
How are fractional solutions shown?
As a reduced fraction together with its decimal approximation. For 3x + 2y = 12 with x − y = 1, the answer appears as x = 14/5 (2.8) and y = 9/5 (1.8), and copying writes the exact fractions.
What happens with no solution or infinitely many?
If the coefficient ratios match but the constants do not, it reports "no solution"; if every ratio matches, it reports "infinitely many solutions". In 2D these correspond to parallel lines and the identical line.
Can I find the intersection of two lines?
Yes. Rewrite y = 2x − 1 and y = −x + 5 as −2x + y = −1 and x + y = 5; entering those gives the intersection point (2, 3).
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Verified: Known values ((2,1), (2,3), fractional 14/5 and 9/5, 3-variable (1,2,3) cross-checked against Cramer's rule), no-solution, infinite, zero-column, row-swap and the English page are covered by browser tests
Did you know?
There is more than one way to solve a system: elimination, substitution and Cramer's rule all reach the same answer. For checking, substitute the solution back into the original equations and confirm both sides match — the most reliable verification.