System of Equations Solver

About the System of Equations Solver

Type two to eight linear equations — one per line, in any arrangement, such as 2x + 3y = 7 and y = 2x − 1 — or fill in the coefficient matrix. The solver works in exact fractions (decimals are converted exactly), so answers like x = 7/5 come out exact rather than as 1.3999999.

It tells you whether the system has one solution, no solution or infinitely many, using the ranks of the coefficient matrix A and the augmented matrix [A | b]. For dependent systems it gives the general solution in parametric and vector form, such as x = 3 − 2t, y = t. Choose how to see the working: substitution, elimination, Cramer’s rule, Gaussian elimination, Gauss–Jordan elimination or the inverse matrix. Every unique solution is checked in the original equations, and a system in two unknowns is drawn as lines that meet at the solution.

Common uses

• Checking homework on pairs of linear equations (Class 10) and three-variable systems, with the method your teacher asks for.
• Solving word problems — mixtures, ages, prices, currents in a circuit — once they are written as equations.
• Seeing why a system has no solution or infinitely many, and writing the general solution.
• Row-reducing an augmented matrix or finding a 2 × 2 or 3 × 3 inverse as part of a linear algebra exercise.

One, none or infinitely many solutions

Gauss–Jordan elimination reduces the augmented matrix [A | b] to reduced row echelon form. If a row becomes 0 = c with c ≠ 0, the equations contradict each other and there is no solution — then rank(A) is less than rank([A | b]). Otherwise rank(A) = rank([A | b]) (the Rouché–Capelli theorem): if this rank equals the number of unknowns there is exactly one solution; if it is smaller, the unknowns without a leading 1 are free parameters and there are infinitely many solutions. Gaussian elimination itself is described in Golub & Van Loan, Matrix Computations (4th ed., 2013), chapter 3.

The methods

• Substitution: solve one equation for one unknown (one with coefficient ±1 when possible), substitute it into the others, repeat, then back-substitute.
• Elimination: multiply two equations by whole numbers so that one unknown has opposite coefficients, then add or subtract to remove it.
• Gaussian elimination: use row operations on the augmented matrix to reach row echelon form, then back-substitute from the last row.
• Gauss–Jordan elimination: continue until each leading entry is 1 and the only non-zero entry in its column, then read off the answer.
• Cramer’s rule: for n equations in n unknowns with D = det A ≠ 0, each unknown is xᵢ = Dᵢ/D, where Dᵢ is the determinant with column i replaced by the constants.
• Inverse matrix: write the system as AX = B; if det A ≠ 0 then X = A⁻¹B. A 2 × 2 inverse is (1/D)·[d −b; −c a]; larger inverses are found by row-reducing [A | I] to [I | A⁻¹].

Typing the equations

• One equation per line, or separate them with semicolons: 2x + 3y = 7; x − y = 1.
• Terms may be on either side and in brackets: 2(x + y) = 10, y = 2x − 1.
• Coefficients can be fractions or decimals: x/2 + 0.25y = 3.
• Unknowns can be any letters, or numbered: x1 + 2x2 − x3 = 4 (also x_1 or x₁).

Limitations

• Linear equations only: each unknown appears to the first power and is not multiplied by another unknown. Equations such as xy = 2 or x² + y = 3 are rejected with an explanation.
• Two to eight equations and up to eight unknowns. Coefficients must be rational numbers (no √2 or π) of up to 60 digits each.
• Cramer’s rule and the inverse matrix need a square system with a non-zero determinant; for other systems the page says so and Gaussian elimination gives the answer.

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