System of Equations Calculator
Enter the coefficients of two or three linear equations to find where the lines or planes intersect. Switch between a 2x2 and a 3x3 system, choose Cramer's rule or Gaussian elimination, request exact fractions, and read every arithmetic step in plain English.
Formula
Worked example
2x + 3y = 6 and 4x - y = 5: D = (2)(-1) - (3)(4) = -14. Dx = (6)(-1) - (3)(5) = -21. Dy = (2)(5) - (6)(4) = -14. x = -21/-14 = 3/2, y = -14/-14 = 1. Check: 2(1.5)+3(1) = 6 and 4(1.5)-1 = 5. Both balance.
How Cramer's rule solves a 2x2 system
Cramer's rule converts solving two linear equations into computing three small determinants. The main determinant D = a1*b2 - b1*a2 comes from the coefficient matrix. To find x, replace the x-column with the constants c1 and c2 and compute that determinant Dx, then divide by D. To find y, replace the y-column with the constants to get Dy and again divide by D. As long as D is non-zero this gives the exact coordinates of the point where the two lines intersect, with no rearranging or substitution.
Extending to a 3x3 system
A 3x3 system has three equations and three unknowns x, y, and z. The coefficient matrix is now 3 rows by 3 columns and its determinant D uses the Sarrus rule or cofactor expansion: D = a1(b2*d3 - d2*b3) - b1(a2*d3 - d2*a3) + d1(a2*b3 - b2*a3). Cramer's rule extends naturally: replace the x-column with the constants to get Dx, the y-column to get Dy, and the z-column to get Dz, then divide each by D. The steps panel shows every sub-determinant laid out with your actual numbers.
When there is no unique solution
A unique solution exists only when D is non-zero. If D equals zero the system is degenerate: the equations are either dependent (same line or plane, infinitely many solutions) or inconsistent (parallel lines or planes, no solution). This calculator distinguishes both cases and explains which situation you are facing. For a 3x3 system the degenerate case is harder to classify completely (it may involve a whole line of solutions), so the calculator conservatively reports the singularity.
Setting up equations in standard form
Rearrange every equation into the form ax + by = c (or ax + by + dz = c for 3x3) before entering coefficients, moving all variable terms to the left and the constant to the right. A missing term means a zero coefficient: x - 5 = 2y becomes 1x - 2y = 5, so a = 1, b = -2, c = 5. Negative signs matter, so double-check each row before reading the result. The fraction toggle shows solutions like 3/2 instead of 1.5 when the answer is a simple rational number.
Choosing a solution method
The calculator supports two methods selectable from the drop-down. Cramer's rule is a direct formula using determinants and is ideal for checking work on paper because every sub-determinant has a clear meaning. Gaussian elimination uses row operations on the augmented matrix to reduce it to row-echelon form and then back-substitutes, which is the standard algorithm taught in linear algebra courses and is also how computer algebra systems solve larger systems. Both methods produce the same answer.
Determinant outcomes for linear systems
| System | Determinant D | Geometry | Solutions |
|---|---|---|---|
| 2x2 | D not 0 | Two lines cross at one point | Unique |
| 2x2 | D = 0, both constant dets = 0 | Same line | Infinitely many |
| 2x2 | D = 0, at least one constant det not 0 | Parallel lines | None |
| 3x3 | D not 0 | Three planes meet at one point | Unique |
| 3x3 | D = 0 | Planes are dependent or inconsistent | None or infinite |
What the value of D tells you about the geometry of the equations.
Frequently asked questions
What is the difference between a 2x2 and a 3x3 system?
A 2x2 system has two linear equations and two unknowns (x and y). Geometrically, each equation is a line in the xy-plane and the solution is where they cross. A 3x3 system adds a third equation and a third unknown z. Each equation is now a plane in 3D space and the unique solution is the single point where all three planes meet.
What does it mean when the determinant is zero?
A zero determinant means the coefficient matrix is singular and the lines (or planes) are not independent. For a 2x2 system they are either the same line (infinitely many solutions) or parallel (no solution). The calculator checks both by also computing the constant determinants Dx and Dy.
How do I enter an equation that is missing a variable?
Use zero for the missing coefficient. For example, 3x = 9 has no y-term, so enter a = 3, b = 0, c = 9. Likewise 2y = 8 becomes a = 0, b = 2, c = 8. For a 3x3 system, use d = 0 for any equation that has no z-term.
Is Cramer's rule the same answer as substitution or elimination?
Yes. Substitution, elimination, Gaussian elimination, and Cramer's rule are all exact methods and produce identical answers for a system with a unique solution. Cramer's rule is convenient for 2x2 and 3x3 problems because it is a direct formula, while Gaussian elimination scales better to larger systems.
When should I use exact fractions instead of decimals?
Turn on the fraction toggle when you want to verify a hand-calculated answer or when the solution is a simple rational number like 1/3 or 5/2. The calculator checks denominators up to 1000 and displays a fraction only when the value matches exactly. For irrational or large-denominator results it falls back to the decimal you chose.
How do I check my answer after solving?
The steps panel includes a verification step that substitutes x and y (and z for 3x3) back into every original equation and checks that both sides balance. You can also do this by hand: multiply each coefficient by the solution value and add the products; the sum should equal the constant c on the right side.