What does it mean when a simultaneous equation system has no solution?

A system has no solution (it is called inconsistent) when the equations represent parallel lines (in 2 variables) or parallel planes (in 3 variables) that never intersect. This happens when the determinant of the coefficient matrix equals zero and the constants produce contradictory equations. For example, x + y = 3 and x + y = 5 cannot both be true at the same time. Geometrically, the two lines are parallel and never cross.

What does it mean when a simultaneous equation system has infinitely many solutions?

A system has infinitely many solutions (it is called dependent) when two or more equations are essentially the same equation written differently, meaning the lines or planes coincide. The determinant of the coefficient matrix is zero. For example, x + y = 4 and 2x + 2y = 8 are the same equation multiplied by 2, so every point on the line x + y = 4 is a solution. You would express the solution as y = 4 - x (or a parametric form) rather than a single point.

What is Cramer's rule and when should I use it?

Cramer's rule is a formula that expresses the solution of a linear system in terms of determinants. For a 2x2 system ax + by = e and cx + dy = f, the determinant D = ad - bc. If D is non-zero, then x = (ed - bf) / D and y = (af - ec) / D. Cramer's rule is elegant for 2x2 and 3x3 systems and is straightforward to apply by hand. For larger systems (4x4 and above), Gaussian elimination is more efficient as Cramer's rule becomes computationally expensive. This solver uses Cramer's rule for 2x2 systems and Gaussian elimination with back substitution for 3x3 systems.

Simultaneous Equation Solver

This calculator solves systems of two or three simultaneous linear equations, working out the exact values of x, y and z (or x and y) that satisfy every equation at once. Choose the 2-variable or 3-variable tab, then type in the coefficients and constants for each equation, using negative numbers for subtraction and decimals where needed. For 2x2 systems the solver applies Cramer's rule, calculating the determinant and using it to work out each variable directly. For 3x3 systems it uses Gaussian elimination with partial pivoting, reducing the equations to row echelon form and then back-substituting to find z, then y, then x. Results update live as you type, showing the solution values, a status message confirming whether the system has one unique solution, no solution, meaning the equations describe parallel lines or planes that never meet, or infinitely many solutions, meaning the equations are dependent, and a full step-by-step breakdown of the working, including a check that substitutes the answer back into each original equation. This makes it useful for checking homework, working through NCEA algebra problems, or solving equations that come up in trade, engineering or business calculations without doing the arithmetic by hand. Results are shown to 8 significant figures, so for equations with very large or very small coefficients, always verify the answer by substituting it back into your original equations.

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Standard method Cramer's rule (2x2) and Gaussian elimination (3x3). Results are exact for integer and decimal inputs.

Enter Coefficients

Enter each coefficient and constant. Use negative numbers for subtraction (e.g. -3). Decimals are accepted.

Eq 1:

Eq 2:

Enter your coefficients above.

How Simultaneous Equations Work

A system of simultaneous linear equations is a set of two or more equations that share the same variables and must all be true at the same time. The solution is the set of variable values that satisfies every equation simultaneously. Geometrically, for a 2-variable system, each equation represents a straight line and the solution is the point where those lines intersect.

Methods for Solving

Method	Best for	How it works
Substitution	Small systems, one variable easily isolated	Rearrange one equation to express one variable in terms of the others, then substitute into the remaining equations
Elimination	2x2 and 3x3 systems	Multiply equations by constants and add or subtract to eliminate variables one at a time
Cramer's rule	2x2 and 3x3 systems	Express each variable as a ratio of determinants
Gaussian elimination	Any size system	Use row operations to reduce the matrix to row echelon form, then back-substitute
Matrix inverse	Systems where the matrix is well-conditioned	Express as Ax = b and compute x = A⁻¹b

Cramer's Rule for 2x2 Systems

For the system:

a₁₁x + a₁₂y = b₁
a₂₁x + a₂₂y = b₂

The determinant D = a₁₁ × a₂₂ - a₁₂ × a₂₁.

If D is not zero, the unique solution is:

x = (b₁ × a₂₂ - a₁₂ × b₂) / D
y = (a₁₁ × b₂ - b₁ × a₂₁) / D

If D = 0, the system either has no solution (inconsistent, parallel lines) or infinitely many solutions (dependent, coincident lines).

Gaussian Elimination for 3x3 Systems

Gaussian elimination converts the augmented matrix [A|b] to row echelon form using three types of elementary row operations: swapping two rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another. Once in row echelon form, the system is solved by back substitution, starting from the last equation and working upward. This solver uses partial pivoting (choosing the row with the largest leading coefficient at each step) to improve numerical stability.

Worked Example (2x2 default)

System: 2x + y = 5 and x + 3y = 10.

Determinant D = (2)(3) - (1)(1) = 6 - 1 = 5.

x = (5 × 3 - 1 × 10) / 5 = (15 - 10) / 5 = 5 / 5 = 1.

y = (2 × 10 - 5 × 1) / 5 = (20 - 5) / 5 = 15 / 5 = 3.

Check: 2(1) + 3 = 5 ✓ and 1 + 3(3) = 10 ✓.

Worked Example (3x3 default)

System: 2x + y - z = 8, -3x - y + 2z = -11, -2x + y + 2z = -3.

Using Gaussian elimination: x = 2, y = 3, z = -1.

Check: 2(2)+3-(-1) = 8 ✓, -3(2)-3+2(-1) = -11 ✓, -2(2)+3+2(-1) = -3 ✓.

Related Calculators

Sources and method: Cramer's rule and Gaussian elimination with partial pivoting. Standard linear algebra as presented in: Anton, H. and Rorres, C., Elementary Linear Algebra, Wiley; and Stewart, J., Calculus, Cengage. Consistent with NCEA Level 2 and Level 3 Mathematics (algebra strand).

This solver handles systems with exact rational and decimal coefficients. Results are displayed to 8 significant figures. For very large or very small coefficients, floating-point precision may affect the last few decimal places. Always verify solutions by substituting back into the original equations.