System of Equations Calculator — Step-by-Step Solutions

The System of Equations Calculator solves systems of two or three linear equations simultaneously. It uses multiple methods including elimination, substitution, Cramer's rule, and matrix inversion to find the values of all unknowns.

Enter the coefficients of your equations and get instant solutions with detailed step-by-step working. The calculator also classifies each system as having a unique solution, no solution, or infinitely many solutions.

Enter coefficients for the system: a₁x + b₁y = c₁, a₂x + b₂y = c₂

Equation 1

a\u2081

b\u2081

c\u2081

Equation 2

a\u2082

b\u2082

c\u2082

Your calculations will appear here

How to Use the System of Equations Calculator

Choose a mode using the tabs at the top. The four modes are: 2x2 System (two equations, two unknowns), 3x3 System (three equations, three unknowns), Substitution Method (step-by-step substitution for 2x2), and Matrix Method (matrix inverse for 2x2).
Enter the coefficients for each equation. For a 2x2 system, enter the values a, b, and c for each equation in the form ax + by = c. For a 3x3 system, enter a, b, c, and d for each equation in the form ax + by + cz = d.
The formatted equations are displayed below the input fields so you can verify what you have entered. Results appear instantly as you fill in all fields.
Review the step-by-step working to understand the solution method. The calculator shows the determinant check, Cramer's rule application, and verification of the solution.
Use the share button to copy a link to your calculation for saving or sending to others.

Formula and Theory

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The solution is the set of values that satisfies all equations simultaneously. Systems arise frequently in science, engineering, economics, and everyday problem solving.

A 2x2 system has three possible outcomes: a unique solution (the two lines intersect at exactly one point), no solution (the lines are parallel and never meet), or infinitely many solutions (the lines coincide). The determinant of the coefficient matrix determines which case applies: if the determinant is non-zero, there is a unique solution.

Cramer's rule provides a direct formula for the solution of a system of linear equations using determinants. For a 2x2 system ax + by = e, cx + dy = f, the solution is x = (ed - bf) / (ad - bc) and y = (af - ce) / (ad - bc), provided the determinant ad - bc is not zero.

Gaussian elimination is a systematic method for solving systems of any size. It transforms the augmented matrix into row echelon form using elementary row operations: swapping rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another. Back substitution then yields the solution.

The matrix method writes the system as AX = B and solves X = A^(-1)B, where A^(-1) is the inverse of the coefficient matrix. This method is equivalent to Cramer's rule for small systems and is particularly useful when solving multiple systems with the same coefficient matrix.

Worked Examples

Example 1: A simple 2x2 system

Problem: Solve: 2x + 3y = 12 and x - y = 1.

Solution: D = 2(-1) - 1(3) = -5. D_x = 12(-1) - 1(3) = -15. D_y = 2(1) - 12(1) = -10. x = -15/-5 = 3, y = -10/-5 = 2.

Answer: x = 3, y = 2

Example 2: A system with no solution

Problem: Solve: x + 2y = 4 and 2x + 4y = 5.

Solution: D = 1(4) - 2(2) = 0. Since the determinant is zero and the equations are not proportional (4/1 is not equal to 5/2), the system is inconsistent.

Answer: No solution. The lines are parallel.

Example 3: A 3x3 system

Problem: Solve: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2.

Solution: Calculate D = 1(-1*-1 - 1*2) - 1(2*-1 - 1*1) + 1(2*2 - (-1)*1) = -1 + 3 + 5 = 7. Apply Cramer's rule for D_x, D_y, D_z, then divide by D.

Answer: x = 1, y = 2, z = 3

Example 4: Infinitely many solutions

Problem: Solve: x + 2y = 6 and 2x + 4y = 12.

Solution: D = 1(4) - 2(2) = 0. The second equation is exactly twice the first, so the equations are dependent.

Answer: Infinitely many solutions. Any point on the line x + 2y = 6 is a solution.

Frequently Asked Questions

What is the difference between this and the Solve For X calculator?

The Solve For X calculator solves a single equation for one unknown. This calculator solves systems of multiple equations with multiple unknowns simultaneously, finding values that satisfy all equations at once.

How do I know if my system has no solution?

A system has no solution when the equations are inconsistent, meaning they represent parallel lines (2D) or parallel planes (3D) that never intersect. The calculator checks for this by examining the determinant and ranks of the coefficient and augmented matrices.

What is Cramer's rule?

Cramer's rule is a formula that uses determinants to solve a system of linear equations. For each variable, you replace the corresponding column of the coefficient matrix with the constants vector, compute the determinant of the resulting matrix, and divide by the determinant of the original coefficient matrix.

Can I solve systems with more than 3 unknowns?

This calculator supports systems with 2 or 3 unknowns. For larger systems, the same methods (Gaussian elimination, Cramer's rule) apply in theory, but the computations become significantly more involved.

What is the substitution method?

The substitution method solves one equation for one variable, then substitutes that expression into the other equation. This reduces the system to a single equation in one variable, which can be solved directly. The value is then substituted back to find the other variable.