System of Linear Equations Calculator
Solve simultaneous 2x2 and 3x3 systems of linear equations using Cramer’s Rule and matrix determinants with step-by-step work.
Did this calculation save you time?
Student ProjectHi! I'm a student developer building Calculat in my spare time. I was tired of searching for basic math tools and having to click through 10 spammy popups, loan ads, and cookie trackers.
I keep this website 100% free, private, and ad-free. If this helped you with your homework, project, or finances today, bookmarking this page or telling a friend helps me keep building more free tools!
What Is a System of Linear Equations Calculator?
A system of linear equations is a collection of two or more linear equations involving the same set of variables. A solution is an assignment of values to the variables that satisfies all equations simultaneously.
How to Use This Calculator
- Select 2×2 (two variables x and y) or 3×3 (three variables x, y, and z).
- Input the numeric coefficients and constants for each equation.
- View the exact solution vector along with determinant values and Cramer’s Rule steps.
Cramer’s Rule Determinant Formula
x_i = \frac{\det(A_i)}{\det(A)}, \quad \text{where } \det(A) \neq 0Each variable x_i is calculated as the ratio of the determinant of matrix A with column i replaced by the constants vector, divided by the determinant of the coefficient matrix A.
Worked Example
Scenario: Solve: 2x + y = 7 and x + 3y = 11
Coefficient determinant: D = (2)(3) - (1)(1) = 6 - 1 = 5
x-determinant: Dx = (7)(3) - (11)(1) = 21 - 11 = 10
y-determinant: Dy = (2)(11) - (1)(7) = 22 - 7 = 15
Solutions: x = Dx / D = 10 / 5 = 2, y = Dy / D = 15 / 5 = 3
Tips & Key Notes
- If determinant D = 0 and all numerator determinants are 0, the system has infinitely many solutions (dependent system).
- If D = 0 but at least one numerator determinant is non-zero, the lines/planes are parallel and no solution exists (inconsistent system).
- Check solutions by substituting the values into both original equations.
Frequently Asked Questions
What is Cramer’s Rule?
Cramer’s Rule is an explicit formula for solving a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution (D ≠ 0).
What does it mean if determinant D is 0?
If D = 0, the system does not have a unique solution; it is either inconsistent (no solutions, parallel lines) or dependent (infinitely many solutions, overlapping lines).
Can this solve 3 variables?
Yes, switch to the 3×3 tab to solve three simultaneous equations with variables x, y, and z.
How do you check simultaneous equation results?
Substitute the values of x and y into each equation and check if both sides evaluate to identical numbers.
Related Calculators
Explore similar toolsLinear Equation Calculator
Solve linear algebraic equations of the form ax + b = c with full step-by-step mathematical deduction and solution classification.
Quadratic Equation Calculator
Solve quadratic equations ax² + bx + c = 0 using the quadratic formula. Computes discriminant Δ, real and complex conjugate roots, vertex, and axis of symmetry.
Average Calculator
Compute the arithmetic mean, median, mode, sum, count, and range for any set of numbers.