Enter your system
Use standard form: ax + by = c, or ax + by + cz = d.
Download results
Generate a file after calculating a system.
Example data table
| System | Equation 1 | Equation 2 | Equation 3 | Expected result |
|---|---|---|---|---|
| 2×2 unique | 2x + 3y = 13 | 4x − y = 5 | — | x = 2, y = 3 |
| 2×2 infinite | x + 2y = 6 | 2x + 4y = 12 | — | Infinitely many solutions |
| 2×2 none | x + 2y = 6 | 2x + 4y = 9 | — | No solution |
| 3×3 unique | x + y + z = 6 | 2x − y + z = 3 | x + 2y − z = 2 | x = 1, y = 2, z = 3 |
Formula used
For a two-variable system, use ax + by = c and dx + ey = f.
D = ae − bd. When D is not zero, the system has one solution.
x = (ce − bf) / D and y = (af − cd) / D.
These values match the result from eliminating x or y. The calculator also checks each result by substitution. For three variables, it uses row elimination. Each row operation removes a variable while preserving the system.
How to use this calculator
- Choose either a two-variable or three-variable system.
- Enter coefficients in standard form. Enter zero when a term is missing.
- Choose automatic selection or a preferred variable to remove first.
- Set decimal places and select graphing for a two-variable system.
- Press Solve by elimination. Review steps, residuals, and classification.
- Download a CSV or PDF record when needed.
Understanding elimination
Elimination solves a system by removing one variable at a time. The method works well when equations use the same unknowns. First, write every equation in standard form. Keep x and y terms in matching columns. Move constants to the right side. This makes each row easier to compare.
Choose a variable that can be removed cleanly. Multiply one or both equations when needed. The chosen variable must have opposite matching coefficients. Add or subtract the equations next. One unknown disappears. Solve the remaining one-variable equation. Substitute that value into an original equation. Then find the other unknown.
This calculator accepts two-variable and three-variable systems. For two equations, it calculates the determinant and Cramer style cross-check values. It also shows an elimination pathway. For three equations, it uses repeated row elimination. It removes a variable from two rows, then solves the smaller system. Back substitution produces the final coordinates.
A nonzero determinant usually means one unique solution. The graph lines meet once in a two-variable system. A zero determinant needs more attention. Equivalent equations produce infinitely many solutions. Parallel but different equations produce no solution. The calculator checks ranks and residuals to identify these cases.
Use the answer check after every calculation. Substitute each reported value into every original equation. A small residual near zero supports the result. Rounding can create tiny differences. Increase the displayed decimal places when values appear close. Exact fractions are available for integer two-variable inputs.
Elimination is useful in algebra, science, finance, and engineering. It helps compare constraints that act together. Clear formatting prevents common sign errors. Keep multipliers visible. Check each subtraction carefully. Use the graph as a visual confirmation, not as the only proof. Save a CSV or PDF report when you need a record.
Before submitting, inspect coefficient signs and blank fields. A negative constant stays negative after moving terms. Zero coefficients are valid and should be entered as zero. Do not divide by a coefficient before checking whether it is zero. For a three-variable system, use all three equations. Independent equations give the strongest result. Dependent rows may signal redundancy. Inconsistent rows signal a conflict. The worked steps show the row operations. Review them to learn the method rather than only copying the final answer. This improves confidence on future problems.
Frequently asked questions
1. What does elimination do?
Elimination combines scaled equations to cancel one variable. The reduced equation has fewer unknowns. Solve it, then substitute back to find the remaining values.
2. Can I enter decimal coefficients?
Yes. Enter integers, decimals, or negative values. Decimal entries produce decimal solutions. Exact fractions are shown only when the two-variable determinant values are whole numbers.
3. Why is the determinant important?
For two-variable systems, a nonzero determinant confirms a unique solution. A zero determinant means the equations are dependent or inconsistent, so the calculator checks their constants next.
4. What means infinitely many solutions?
The equations describe the same relationship. One equation can be made from another by multiplying every term by the same nonzero factor.
5. What means no solution?
The equations conflict. In two variables, they often represent distinct parallel lines. In row elimination, a contradiction appears, such as zero equaling a nonzero number.
6. How are residuals used?
A residual is the difference between a substituted left side and its constant. Correct answers have residuals of zero, or extremely close to zero after rounding.
7. Can this solve three-variable systems?
Yes. Select the three-equation option. The calculator performs repeated row elimination, determines the rank, and uses back substitution when a unique solution exists.
8. Why might a graph not show?
Graphs are limited to two-variable systems because each equation is a line on a standard x-y plane. Three-variable systems require three-dimensional visualization.
9. Which variable should I eliminate first?
Choose a variable with equal, opposite, or easy-to-match coefficients. The automatic option selects a reasonable path. Either valid elimination path reaches the same unique solution.
10. Does a zero coefficient cause an error?
No. Zero is valid. It means that variable does not appear in that equation. Enter zero rather than leaving the input blank.
11. Can I save my work?
Yes. After solving, download the result as CSV or PDF. The print button also creates a clean paper or browser-PDF copy of the result.