2×2 System Solver (Gaussian Elimination)
Solve a system of 2 linear equations in 2 unknowns with Gaussian elimination. Enter the 2x3 augmented matrix and see every row operation and the final values of x, y.
Solve 2 equations in 2 unknowns. The grid is preset to the 2×3 augmented matrix you need — 2 coefficient columns plus one column of constants.
Reading your system into the grid
Each row of the grid is one equation. Columns 1 to 2 hold the coefficients of x, y, and the final column — separated by the vertical line — holds the constant.
2x + 3y = 8 1x - 1y = -1
Other system sizes
Frequently asked questions
How many operations does it take to solve a 2x2 system? ▾
Gaussian elimination on an 2x2 system needs roughly 5 arithmetic operations — the cost grows with the cube of the size, which is why large systems are solved numerically rather than by hand.
Can a 2x2 system have more than one solution? ▾
Yes. If elimination leaves fewer than 2 pivots, at least one variable is free and the system has infinitely many solutions. If a row reduces to 0 = c with c non-zero, there is no solution at all.
What order do I enter the numbers in? ▾
One row per equation. The first 2 columns hold the coefficients of x, y in order, and the last column holds the constant from the right-hand side of that equation.
Do I need to rearrange my equations first? ▾
Yes — write each equation with all the variables on the left in the same order and the constant on the right before reading off the coefficients. Missing variables get a coefficient of 0.
About this Calculator
Solve a system of 2 linear equations in 2 unknowns with Gaussian elimination. Enter the 2x3 augmented matrix and see every row operation and the final values of x, y.