System of linear equations solver (2×2 and 3×3)
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Result
Solve a system of two or three linear equations. Type the equations as you would write them, such as 2x + y − z = 8, or enter the augmented matrix. The calculator shows every row operation of Gauss–Jordan elimination, or the determinants of Cramer’s rule, works with exact fractions, and recognises systems with no solution or infinitely many solutions, which it writes with a parameter.
Step-by-step solution
The system
Augmented matrix
Column 1 (x): pivot in row 1
Column 2 (y): pivot in row 2
Column 3 (z): pivot in row 3
Read off the solution
Check in the original equations
Key formulas
- 2×2 determinant
- det = ad − bc
- 2×2 inverse
- A⁻¹ = (1/det)·[d −b; −c a]
- Cramer’s rule
- xi = det(Ai)/det(A)
How to solve a system of linear equations by elimination
- Write the system as an augmented matrix: one row per equation, the coefficients of x, y and z, and the right-hand side after the bar.
- Choose a pivot in the first column (a non-zero entry, ideally 1) and swap it into the first row if necessary.
- Divide the pivot row by the pivot, then subtract multiples of it from the other rows to make the rest of the column zero.
- Repeat for the next column and the next row until the coefficient part is in reduced row echelon form.
- Read off the solution. A row 0 = c with c ≠ 0 means there is no solution; a column without a pivot means a free variable and infinitely many solutions.
- Check by substituting the solution into the original equations.
Questions and answers
When should I use Cramer’s rule?
For small systems with a non-zero determinant, especially 2×2. Each unknown is a ratio of determinants: x = Dx/D. For larger systems, or when D = 0, elimination is faster and also tells you whether there are no solutions or infinitely many.
How do I know if a system has no solution?
When elimination produces a row with all coefficients zero but a non-zero right-hand side, such as 0 = 3. Geometrically the lines or planes do not share a common point, for example two parallel lines.
What does infinitely many solutions mean?
One equation is a combination of the others, so there are fewer independent equations than unknowns. The solutions form a line or a plane, written with a parameter t: for example x = 2 − t, y = 1 + t, z = t.
How do I enter the equations?
Separate the equations with semicolons and write the coefficient before each variable: 2x + 3y − z = 5; x − y = 1; 4y + z = 0. Fractions such as 1/2x and decimals are allowed, and a variable can appear on both sides.
What is the rank?
The number of pivots after elimination, which is the number of independent equations. A square system has a unique solution exactly when the rank equals the number of unknowns, which is when the determinant is non-zero.
Practise it yourself
Practise Matrices & determinants
A calculator gives the answer. In the exam you need the method: practise it with new problems, hints and worked solutions.