Algebra
System of Linear Equations Solver
Solve 2×2 and 3×3 linear systems. Solve systems of 2 or 3 linear equations using substitution, elimination, or Cramer's Rule with step-by-step output.
Examples:
Formula Reference
Linear Equation
ax + b = c → x = (c−b)/a
Quadratic Formula
x = (−b ± √(b²−4ac)) / 2a
Discriminant
Δ = b² − 4ac
Vertex Form
y = a(x−h)² + k
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Solution Methodology
01
Enter Coefficients
Input a, b, c values for each equation in the system.
02
Compute Determinant
Evaluate det(A) to classify the system before solving.
03
Apply Elimination
Perform row operations to reach echelon form, then back-substitute for each variable.
Common Questions
What is Cramer's Rule for solving linear systems?
Cramer's Rule expresses each variable as a ratio of determinants. For a 2×2 system ax+by=e, cx+dy=f: x = (ed−bf)/(ad−bc) and y = (af−ec)/(ad−bc). The denominator is the determinant of the coefficient matrix; a zero determinant means the system has no unique solution.
How does Gaussian elimination work?
Gaussian elimination transforms the augmented matrix into row echelon form using three operations: swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another. Back-substitution then extracts the values of each variable from the simplified upper-triangular form.