System of Equations Calculator
Solve a system of linear equations with 2 to 5 unknowns, typed as equations or as a grid of coefficients. Exact fractions, Gauss–Jordan elimination step by step, Cramer’s rule with every determinant, and a clear answer when there is no solution or infinitely many.
How to Use
- Type the equations one per line, such as 2x + y = 5. Fractions (x/2, 1/3), decimals and brackets are fine.
- Or type a grid: one row per equation, the coefficients and then the constant, so 2 1 5 means 2x + y = 5.
- Choose the method: Gauss–Jordan (any size) or Cramer’s rule (2 × 2 and 3 × 3).
- Read the solution, the type (one solution, none or infinitely many), the determinant and the rank.
- With two unknowns the graph shows both lines and where they cross; with three or more it shows the reduced matrix.
- Show Work lists every row operation or determinant, then checks the answer in each equation.
Worked Example
Elimination. For 2x + y = 5 and x − y = 1, add the two equations: the y terms cancel and 3x = 6, so x = 2. Then x − y = 1 gives y = 1. Check: 2·2 + 1 = 5 and 2 − 1 = 1. On the graph, the two lines cross at (2, 1).
Cramer’s rule with fractions. For 3x + 2y = 7 and 4x − 5y = 2, det A = 3·(−5) − 2·4 = −23. Put the constants in the x column: det A₁ = 7·(−5) − 2·2 = −39, so x = −39 ÷ −23 = 39/23. In the y column: det A₂ = 3·2 − 7·4 = −22, so y = 22/23. Rounding to 1.70 and 0.96 part-way through would have given an answer that fails the check.
The common mistake: reading det A = 0 as “no solution”. A zero determinant only says there is no single solution. x + 2y = 4 with 2x + 4y = 5 reduces to 0 = −3, so there is no solution (parallel lines). x + 2y = 4 with 2x + 4y = 8 also has det A = 0, but it reduces to 0 = 0: the same line twice, with infinitely many solutions x = 4 − 2t, y = t.
Show Work
Formulas
From Counting Boards to Gauss–Jordan
The eighth chapter of the Chinese Nine Chapters on the Mathematical Art, compiled by about the first century AD, solved systems of several unknowns by setting the coefficients out in columns on a counting board and eliminating them one at a time, the same procedure taught today. In Europe, Gabriel Cramer published his determinant rule in 1750 in a book on algebraic curves.
Carl Friedrich Gauss used systematic elimination around 1810 to solve the least-squares equations for the orbit of the asteroid Pallas, which is why the method carries his name. In 1888 the geodesist Wilhelm Jordan described the variant that clears the entries above each pivot as well as below, producing the reduced row echelon form that this calculator shows.
About This Calculator
This calculator solves linear systems of up to 6 equations in up to 5 unknowns. It reads ordinary equations (with fractions, decimals and brackets) or a grid of coefficients, works in exact fractions throughout, and explains every step: the row operations of Gauss–Jordan elimination pivot by pivot, or each determinant of Cramer’s rule. When a system has no solution it shows the impossible row; when it has infinitely many it gives the answer in terms of free parameters. For eigenvalues, null spaces and other matrix work, use the Linear Algebra Lab.
Everything runs in your browser; nothing is sent anywhere.
Related tools: Linear Algebra Lab, Matrix Calculator, and Graphing Calculator.
Frequently Asked Questions
How do I solve a system of two equations by elimination?
Add or subtract multiples of the equations so one unknown cancels. For 2x + y = 5 and x − y = 1, adding them gives 3x = 6, so x = 2; putting that back into x − y = 1 gives y = 1. Gauss–Jordan elimination does the same thing in a fixed order on the grid of coefficients, which works for any number of unknowns.
What is Cramer’s rule?
Each unknown is a ratio of two determinants: replace that unknown’s column of coefficients with the constants and divide by the determinant of the coefficients. For 3x + 2y = 7 and 4x − 5y = 2, det A = 3·(−5) − 2·4 = −23, det A₁ = 7·(−5) − 2·2 = −39 and det A₂ = 3·2 − 7·4 = −22, so x = 39/23 and y = 22/23. It only works when det A is not 0.
How can I tell if there is no solution or infinitely many?
Reduce the system. A row that reads 0 = a non-zero number means no solution: x + 2y = 4 and 2x + 4y = 5 reduce to 0 = −3, two parallel lines. A row of all zeros (0 = 0) with fewer pivots than unknowns means infinitely many: x + 2y = 4 and 2x + 4y = 8 are the same line, x = 4 − 2t, y = t for any t.
What is reduced row echelon form (RREF)?
The grid after Gauss–Jordan elimination: each leading entry is 1 and is the only non-zero number in its column. Then the answer can be read straight off. For 2x + y − z = 8, −3x − y + 2z = −11 and −2x + y + 2z = −3 the RREF is [1 0 0 | 2], [0 1 0 | 3], [0 0 1 | −1], so x = 2, y = 3, z = −1.
Can I type fractions and decimals?
Yes, and the answer stays exact. x/2 + y/3 = 4 with x − y = 3 gives x = 6 and y = 3, and 0.2x + 0.5y = 3 with x + y = 10 gives x = 20/3 and y = 10/3 (about 6.667 and 3.333) rather than rounded decimals. Coefficients can also be written with brackets, such as 3(x + 2) − y = 1.
How do I use the System of Equations Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Homework
2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3 gives x = 2, y = 3, z = −1, with every row operation shown.
Mixtures
10 L at 30% from 20% and 50% stock: x + y = 10, 0.2x + 0.5y = 3, so 20/3 L and 10/3 L.
Ticket sales
200 tickets at $12 and $7 taking $2,000: a + c = 200, 12a + 7c = 2000, so 120 adults and 80 children.
Circuits
Mesh currents 10I₁ − 4I₂ = 12 and −4I₁ + 8I₂ = 0 give I₁ = 1.5 A and I₂ = 0.75 A.
Checking an answer
Every solution is substituted back into each equation, so a sign slip in your own working shows up.
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