Solve 2×2 and 3×3 systems of linear equations by elimination. Step-by-step multiply-and-add operations, presets for common systems, elimination steps table, and solution coordinate visual.
The elimination method (also called the addition method) is a systematic technique for solving systems of linear equations. The idea is to multiply one or both equations by constants so that when you add or subtract the equations, one variable cancels out. For a 2×2 system like { 2x + 3y = 7, 4x − 3y = 5 }, adding the equations directly eliminates y: 6x = 12, so x = 2, then back-substitution gives y = 1. For 3×3 systems, the process extends to Gaussian elimination — reducing the system to an upper triangular form by successively eliminating variables. This calculator handles both 2×2 and 3×3 systems, showing every multiply and add step in a detailed table. It detects special cases: no solution (parallel lines, inconsistent), infinite solutions (dependent/coincident), and unique solutions. Presets include classic textbook systems, including ones with no or infinite solutions. A coordinate grid visual plots the solution point (for 2×2 systems) at the intersection of the two lines.
The elimination method is a core algebraic technique, but students often make sign errors when multiplying and subtracting equations. This calculator traces every step — showing which multiplier is applied to which equation, the intermediate row after each elimination, and the final back-substitution. Presets include systems with unique, infinite, and no solutions, making it a complete learning tool for linear algebra.
2×2: a₁x + b₁y = c₁, a₂x + b₂y = c₂ Multiply eq1 by a₂, eq2 by a₁, subtract → solve for y Back-substitute to find x 3×3: Gaussian forward elimination + back-substitution
Result: x = 2, y = 1
Add: 2x+3y=7 + 4x−3y=5 → 6x=12 → x=2. Substitute: 2(2)+3y=7 → y=1.
Elimination, substitution, and matrix methods (Cramer's rule, inverse matrices) are three approaches to solving linear systems. Elimination is often the most efficient for 2×2 and 3×3 systems by hand because it avoids fractions when coefficients already have matching magnitudes. Substitution is better when one variable is already isolated (e.g., y = 3x + 1). For larger systems (4+ unknowns), matrix methods and computational tools are preferred.
A 2×2 system has no solution when the lines are parallel (same slope, different intercepts) — corresponding to a zero determinant with an inconsistent constant. It has infinitely many solutions when the equations are multiples of each other (same line). The elimination method detects both: if both variables cancel and you get 0 = nonzero, there is no solution; if you get 0 = 0, there are infinitely many. The determinant D = a₁b₂ − a₂b₁ being zero is the algebraic indicator.
Elimination is the manual version of Gaussian row reduction on the augmented matrix [A|b]. Each "multiply equation 1 by k and add to equation 2" corresponds to the elementary row operation R₂ ← R₂ + kR₁. The goal is to reach row echelon form (upper triangular), then back-substitute. This connection makes elimination the conceptual bridge between basic algebra and linear algebra courses.
A technique where you multiply equations by constants and add/subtract them to eliminate one variable, making the system easier to solve. Use this as a practical reminder before finalizing the result.
Elimination is best when coefficients can be easily matched. Substitution is better when one variable is already isolated.
If you get 0 = 0, the system has infinite solutions (dependent). If you get 0 = nonzero, there is no solution (inconsistent).
Yes — extend to Gaussian elimination: eliminate one variable from two pairs of equations to get a 2×2 system, then solve that. Keep this note short and outcome-focused for reuse.
A systematic version of the elimination method that reduces any system to row echelon (upper triangular) form by eliminating variables column by column. Apply this check where your workflow is most sensitive.
Yes — elimination is equivalent to row reduction of the augmented matrix. Each elimination step corresponds to an elementary row operation.