System of Equations Calculator
System of Equations Calculator
Solve two or three linear equations at once. Type the coefficients and get exact answers as fractions, the elimination worked out row by row, a Cramer’s rule check, a graph of the lines, and a plain answer when there is no solution or infinitely many.
The lines on a graph
Row reduction
The system is solved by Gauss Jordan elimination with exact fractions, so answers such as 11/5 are exact rather than rounded, and decimals you type are turned into fractions first. The determinant decides the type of answer: if it is not zero there is exactly one solution; if it is zero the equations either contradict each other or overlap, giving no solution or infinitely many. Cramer’s rule is shown as an independent check. The CSV, image and printed report contain the values you entered, nothing preloaded.
How to use this tool
Choose two or three unknowns, then type the coefficients of each equation into its row: the number in front of x, the number in front of y, and, for three unknowns, z, with the plain number on the right after the equals sign. A blank box counts as 0, and fractions such as 3/4 or decimals such as 0.25 are fine. The answer, the working and the graph update as you type.
The entry people most often get wrong is the arrangement. Each equation must have every unknown on the left and a single number on the right. For y = 2x − 3, move the 2x across first to get −2x + y = −3, then type −2, 1 and −3.
What the calculation is doing
The calculator writes the system as a grid of numbers, the augmented matrix, and applies three legal moves: swap two rows, scale a row, and add a multiple of one row to another. None of them changes the solution, because each one turns true equations into other true equations. The aim is 1s down the diagonal and 0s elsewhere, at which point each row reads one unknown directly.
All arithmetic is done with exact fractions, so 2x + 3y = 8 and x − y = 1 comes out as x = 11/5 and y = 6/5 rather than 2.2000000001. As an independent check, Cramer’s rule solves the same system with determinants and must agree.
Why a system can have one, none or infinitely many answers
Each linear equation in two unknowns is a straight line, and two lines can do only three things: cross once, run parallel, or lie on top of each other. The determinant, ad − bc for the coefficients a, b, c and d, measures how far the two lines are from being parallel. If it is not zero they cross at exactly one point. If it is zero they have the same slope, and the right hand sides decide whether they are separate parallel lines, with no solution, or the same line, with infinitely many.
The same logic holds with three unknowns, where each equation is a flat plane in space. Three planes usually meet at a single point, but they can also share a whole line, or fail to have any point in common. Elimination finds out which by what the last rows reduce to: a clean 1 on the diagonal, a row of zeros, or the impossible 0 = 1.
Methods compared
| Method | Idea | Best for | Watch out |
|---|---|---|---|
| Elimination | Add multiples of one equation to another to cancel a variable | Any size; the method used by computers | Hand working gets long past 3 unknowns |
| Substitution | Solve one equation for a variable and put it into the others | 2 unknowns with a coefficient of 1 somewhere | Fractions pile up quickly |
| Cramer’s rule | Each unknown is a ratio of two determinants | 2 or 3 unknowns, and checking answers | Fails when the determinant is 0 |
| Graphing | Draw each line and read off the crossing | Seeing why there is no solution or many | Only as accurate as the drawing |
| Matrix inverse | x = A inverse times b | Many systems with the same left side | Same limits as Cramer’s rule |
A worked example you can check on paper
Take the starting example: 2x + 3y = 12 and x − y = 1. Multiply the second equation by 3 so the y terms will cancel: 3x − 3y = 3. Add it to the first: 5x = 15, so x = 3. Put x = 3 back into x − y = 1, and y = 2.
Cramer’s rule gives the same answer. The determinant is 2 x (−1) − 3 x 1 = −5. Replacing the x column with the right hand sides gives 12 x (−1) − 3 x 1 = −15, so x = −15 ÷ −5 = 3. Replacing the y column gives 2 x 1 − 12 x 1 = −10, so y = −10 ÷ −5 = 2. Check in the first equation: 2 x 3 + 3 x 2 = 12.
Mistakes that catch people out
Leaving a variable on the right. Every unknown must be on the left side before the coefficients are typed in, or the system solved is a different one.
Dropping a minus sign. x − y = 1 has a y coefficient of −1, not 1. Most wrong answers trace back to one sign copied wrongly.
Forgetting a missing variable is a 0. In x + z = 5 the y coefficient is 0. Leave its box empty or type 0.
Multiplying only one side. Scaling an equation means multiplying every term, including the number on the right.
Reading a zero determinant as x = 0. It means there is no single answer, not that the answer is zero. Look at the reduced rows to see whether there are none or infinitely many.
Popular calculations
Each opens the calculator with those equations filled in, and they can be changed once it is open.
Two equations with one solution
No solution or infinitely many
Three equations in x, y and z
How this was checked
The elimination uses exact fractions throughout, and every solution was substituted back into each original equation, as the check table under the calculator does. Every unique answer was also solved independently by Cramer’s rule and the two methods agreed exactly. Systems with no solution and with infinitely many were checked against their reduced rows, and the parametric answers were substituted back for several values of t.
Every address in the lists above was then opened and checked for a distinct title, the right answer, a graph and matrix that render, and clean CSV, image and print output containing only that system, at desktop, tablet and phone widths.
Common questions
How do you solve a system of two equations?
Eliminate one variable. For 2x + 3y = 12 and x − y = 1, multiply the second by 3 to get 3x − 3y = 3 and add it to the first: 5x = 15, so x = 3. Put x = 3 back into x − y = 1 to get y = 2.
How do you know if a system has no solution?
When elimination produces a row that reads 0 = a nonzero number, the equations contradict each other. For two equations in x and y it means the lines are parallel, which happens when the determinant is 0 but the equations are not multiples of each other.
What does infinitely many solutions mean?
One equation adds no new information because it is a multiple or combination of the others, such as x + 2y = 3 and 2x + 4y = 6. Every point on the shared line is a solution, written as x = 3 − 2t, y = t for any t.
What is Cramer’s rule?
For a system with a nonzero determinant D, each unknown equals a second determinant divided by D. The second determinant is D with that unknown’s column replaced by the right hand sides. For 2x + 3y = 12 and x − y = 1, D = −5 and x = −15 ÷ −5 = 3.
Can I enter fractions and decimals?
Yes. Type 3/4, −2/3 or 0.25 in any box. Decimals are converted to exact fractions, 0.25 becomes 1/4, and the answer is given exactly, with a decimal version alongside when it is not a whole number.
How do I solve three equations with three unknowns?
Choose 3 equations, then eliminate x from the second and third using the first, eliminate y from the third using the second, and back substitute. The calculator shows each of these row operations and the matrix after every step.
Related Calculators and Reference
- Quadratic Formula Calculator: when one equation has an x squared term.
- Slope Calculator: the slope and intercept of each line in a 2 variable system.
- Fraction Calculator: work with the exact fraction answers.
- Scientific Calculator: evaluate the equations at the answer.
- Matrix Calculator: determinants, inverses and products of the coefficient matrix.
- MIT OpenCourseWare 18.06 Linear Algebra: Gilbert Strang’s course, which starts from solving systems by elimination.