System of Equations Calculator — Solve 2 to 5 Unknowns with Steps
System of equations calculator: enter the coefficients of a linear system with 2 to 5 unknowns and get the exact solution as fractions, with every Gauss–Jordan elimination step shown. It also detects no-solution and infinitely-many-solution cases and checks the answer.
Worked example: solving a 2-unknown system by elimination
This is the kind of system that often comes from word problems such as "some items cost 10 each and others 20 each…". The core of the elimination method is to multiply one equation, subtract it from the other and make one variable disappear, and the working shown by this tool follows exactly the same idea.
| Equation ① | x + y = 10 |
|---|---|
| Equation ② | 2x + 4y = 26 |
| ② − ①×2 | 2y = 6 → y = 3 |
| Substitute into ① | x + 3 = 10 → x = 7 |
| Solution | x = 7, y = 3 |
What is the System of Equations Calculator?
A system of equations asks for the values of the variables that satisfy two or more equations at the same time. This calculator solves systems of linear equations from two unknowns (x and y) up to five unknowns (x, y, z, w and v) — you simply enter the coefficients and the right-hand sides. Results are shown exactly as fractions rather than rounded decimals, and every intermediate step (which equation was multiplied by what and subtracted from which, as in Gauss–Jordan elimination) is listed, so you can use it to check answers you worked out by hand or to pinpoint where a calculation went wrong.
A system of linear equations always ends up in one of three situations: "exactly one solution", "no solution (inconsistent)" or "infinitely many solutions (dependent)". For example, x + y = 2 and 2x + 2y = 5 have proportional left-hand sides but right-hand sides that do not match, so there is no solution, while x + y = 2 and 2x + 2y = 4 are essentially the same equation, so there are infinitely many solutions. The tool makes this judgement automatically and, in the dependent case, shows the general solution using parameters such as t. It only handles linear equations (no squared variables and no products of variables), so systems containing terms like x² or xy are not supported.
How to use the System of Equations Calculator
- Choose the number of unknowns Pick anything from "2 unknowns" to "5 unknowns". The number of equations matches the number of unknowns, and the variables are used in the order x, y, z, w, v.
- Rewrite each equation as ax + by = c Move the terms so that the variable terms are on the left and the plain numbers are on the right. For example, rewrite 3x = 5 − 2y as 3x + 2y = 5 before entering it.
- Enter the coefficients and right-hand sides For subtracted terms, use a negative coefficient (for x − 2y, the coefficient of y is −2), and enter 0 for any variable that does not appear. Fractions can be entered as 3/4 and decimals as 0.5.
- Check the solution and the verification As soon as every field is filled in, the solution appears instantly. The value of each left-hand side with the solution substituted is also listed, so you can confirm that it matches the right-hand side.
- Follow the working The "Working" section below shows each step in order — swapping rows, multiplying a row, or subtracting a multiple of one row from another — together with the coefficients right after each step.
Tips for getting more out of it
- When the coefficients contain fractions, it is easier to multiply both sides by the least common multiple of the denominators first so that everything becomes an integer. Of course, entering fractions such as 1/2 directly gives exactly the same solution.
- Decimals such as 0.1 are handled internally as the fraction 1/10, so floating-point rounding errors never shift the answer.
- The working proceeds column by column from the left, leaving the coefficient of each column in just one equation. The order in which variables are eliminated may differ from a textbook elimination, but the final solution is the same.
- For two-unknown problems where you want to see the intersection on a graph, draw both lines with the linear function grapher and you can confirm visually that the coordinates of the intersection point are the solution of the system.
When the System of Equations Calculator comes in handy
Checking math homework and test answers
Systems of equations are a standard topic in secondary school algebra. Because the working is shown step by step, you can find exactly where a coefficient went wrong when your answer does not match.
Solving word problems about quantities and prices
Problems such as "You bought 10 apples and oranges in total and paid 1,000 yen" — where you work out the breakdown from a total count and a total amount — can be turned straight into equations and solved.
Solving systems with three or more unknowns
Problems with three or more unknowns, such as those in high school and university linear algebra or circuit analysis with Kirchhoff's laws, are tedious and error-prone by hand, which makes this tool ideal for checking your work.
Finding mixing and blending ratios
When you want to mix salt solutions of different concentrations to reach a target concentration and amount, you can set up a system of equations to find how many grams of each to use.
Checking the chickens and rabbits puzzle with equations
The classic heads-and-legs arithmetic puzzle about chickens and rabbits (known in Japan as cranes and turtles, tsurukame-zan) has exactly the same structure as a two-unknown system. You can see the area-diagram approach in the chickens and rabbits problem calculator.
System of equations glossary
- System of linear equations
- A set of several equations that are linear in the variables (no squares and no products of variables). A set of values that satisfies all the equations at the same time is called a solution.
- Number of unknowns
- How many variables you are solving for. A system for x and y has two unknowns, and a system for x, y and z has three unknowns.
- Elimination method
- A method that multiplies equations by suitable numbers and adds or subtracts them to eliminate one variable at a time. It is usually the first method students learn, and it forms the basis of Gauss–Jordan elimination.
- Substitution method
- A method that rearranges one equation into the form "x = …" and substitutes it into the other equation to reduce the number of variables. It is quick when a variable has a coefficient of 1.
- Gauss–Jordan elimination
- A method that arranges the coefficients as a matrix and repeatedly swaps rows, multiplies rows by constants and adds or subtracts multiples of other rows, until each column's coefficient remains in only one row. The working in this tool is displayed using this method.
- Augmented matrix
- The coefficient matrix of a system with one extra column for the right-hand sides added on the right. It lets you follow the transformations using numbers alone, without writing out the equations.
- Inconsistent / dependent
- A system is inconsistent when it has no solution at all, and dependent when it has infinitely many solutions and no single solution can be determined.
- Rank
- The number of nonzero rows that remain after Gauss–Jordan elimination. If the ranks of the coefficient matrix and the augmented matrix differ, the system is inconsistent; if they agree but are smaller than the number of unknowns, the system is dependent.
Frequently Asked Questions
Side Note — From China 2,000 years ago to Seki Takakazu, and on to Gauss
The idea of solving systems of linear equations using only the array of coefficients has been known for a surprisingly long time. Chapter 8, "Fangcheng", of the Chinese mathematical text The Nine Chapters on the Mathematical Art (thought to have been completed by around the 1st century AD) describes arranging counting rods in vertical columns on a board and eliminating unknowns by multiplying columns and subtracting them from one another. This is almost the same idea as today's Gauss–Jordan elimination, and the text even includes rules that assume negative numbers will appear partway through the calculation.
In Japan, the Edo-period mathematician Seki Takakazu, in his 1683 work Kai Fukudai no Hō, organised the expressions that arise while eliminating variables from equations and dealt with a concept corresponding to what we now call the determinant. His treatment contained errors for orders of four and above, but Leibniz in Europe wrote down a similar idea in a letter only in 1693 (some argue his drafts date from around 1678), which means Seki reached this idea independently of Europe, at about the same time or even earlier.
The name "Gaussian elimination" comes from the fact that in the early 19th century the mathematician Gauss, while solving least-squares systems for problems such as computing the orbit of the asteroid Pallas, devised his own notation for organising the elimination steps, and this notation later spread among professional computers. Gauss himself, however, called the method "common elimination" (eliminatio vulgaris), and he was not its inventor. Even as of 1923 it was not yet called by Gauss's name; the term only became established in the 20th century.
Today, systems of equations with thousands or tens of thousands of unknowns are routinely solved by computers in weather forecasting, structural strength calculations, image processing and many other fields. Yet the idea at their core — eliminating variables one at a time — has not changed one bit from the elimination method taught in school.