Welcome to the system of equations calculator, where we will learn how to solve a system of linear equations. Our handy calculator will quickly find the solution to any problem you give it, and if there are two equations and an infinite number of solutions, it will even tell you what they look like! The system of equations solver uses the so-called Gaussian elimination method, but this is not the only method, so below we present several answers to the question "How to solve a system of equations?"
Let's not waste a second longer and get to it, shall we?
What is a system of linear equations?
Remember all those riddles on social media — you know, the ones where three apples are equal to 30, an apple and two bananas are equal to 18, and a banana minus a coconut equals two, and you had to calculate how much the apple, banana, and coconut are worth? That is what mathematicians call a system of linear equations.
"But how? Mathematicians don't use apples and bananas, do they?" Well, they too like to keep the doctor away and bite into an apple from time to time, but you're right — they don't calculate in apples. However, it doesn't make any difference whether you write "three apples equal 30," or 3x = 30.
The x that appeared above is what we call a variable. It denotes a number or element whose value we don't know, but about which we do know something. In our case, we know that three apples equal 30, but the apple is simply a variable, like x, as we don't know its value. In essence, "what is the solution to the system of equations…" is the same as "give me the value of an apple (or x) that satisfies…"
"But what does linear mean?" We say that an equation is linear if its variables (be they x's or coconuts) are to the first power. This means that they are not squared (x²) as in quadratic equations, nor in the denominator of a fraction, nor under a square root. They can, however, be multiplied by any number, just as we had the 3 in our 3x = 30 equation. For example, the equation -2x + 14y - 0.3z = 0 is linear, but 10x - 7y + z² = 1 isn't.
💡 Check our quadratic formula calculator to understand what a quadratic equation is!
Lastly, if we have a few equations to solve together, then we call them a system of equations. We denote it by drawing a curly bracket to the left of them. This means that we're only interested in solutions to all the equations in the system. If we find values that work for the first equation but don't for the second, then we don't call that a solution.
How to solve a system of equations?
There are several classic methods to solve a system of linear equations. Here are the five most popular ones:
- Substitution — solve one equation for one variable, then substitute that expression into the other equations.
- Elimination (addition method) — add or subtract multiples of the equations to cancel a variable.
- Gaussian elimination — a systematic version of elimination performed on the augmented matrix (this is the method our calculator uses).
- Cramer's rule — uses determinants; works only when the coefficient matrix is square and its determinant is not zero.
- Graphing — plot each equation as a line (or plane) and read off the intersection point.
Every system of linear equations has exactly one of three possible outcomes:
- Exactly one solution — the lines (or planes) meet at a single point. The coefficient determinant is non-zero.
- No solution — the system is inconsistent; the lines are parallel and never meet.
- Infinitely many solutions — the equations describe the same line (or intersecting planes), so there is a whole family of solutions, which our calculator writes using a free parameter.
Example: Using the system of equations solver
Say we want to solve the system:
2x + 3y = 8x - y = -1
- Choose 2 variables (2×2) as the system size.
- Enter the coefficients of the first equation:
2,3, and the right-hand side8. - Enter the coefficients of the second equation:
1,-1, and the right-hand side-1. - Read the answer:
x = 1andy = 2. The calculator also shows the augmented matrix, its reduced row echelon form, the determinant, and a substitution check.
Example: solving systems of equations by Gaussian elimination
Gaussian elimination writes the system as an augmented matrix [A | b] and then uses three elementary row operations — swapping two rows, multiplying a row by a non-zero number, and adding a multiple of one row to another — until the matrix reaches reduced row echelon form (RREF). Once in RREF, the solution can simply be read off.
For the 3×3 system x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2, elimination reduces the augmented matrix to the identity on the left, giving x = 1, y = 2, z = 3. Our calculator performs exactly these steps and displays both the starting matrix and the final RREF so you can follow along.
Frequently Asked Questions
What does it mean when a system has no solution?
It means the system is inconsistent — no set of values can satisfy all equations at once. Geometrically, the lines are parallel (2D) or the planes have no common point (3D). During Gaussian elimination this shows up as a row of the form 0 = c where c is not zero.
When does a system have infinitely many solutions?
When the equations are not independent — one can be obtained from the others. The rank of the coefficient matrix is smaller than the number of variables, so at least one variable is free. Our calculator expresses the other variables in terms of a parameter such as t.
What is the determinant used for?
For a square system, if the determinant of the coefficient matrix is non-zero, there is exactly one solution (and Cramer's rule applies). If the determinant is zero, the system has either no solution or infinitely many.
Can I enter fractions or decimals?
Yes. You can type decimals like 0.5, negative numbers like -3, or simple fractions like 3/4 in any coefficient field.
Does the unit system change the math?
No — the underlying algebra is identical. The unit selector (metric such as meters or kilograms, or US customary such as feet or pounds) simply labels the solved values so the answer reads in the units your problem is stated in.