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
Equation 2
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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.
Problema: Solve: 2x + 3y = 12 and x - y = 1.
Solucion: 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.
Respuesta: x = 3, y = 2
Problema: Solve: x + 2y = 4 and 2x + 4y = 5.
Solucion: 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.
Respuesta: No solution. The lines are parallel.
Problema: Solve: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2.
Solucion: 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.
Respuesta: x = 1, y = 2, z = 3
Problema: Solve: x + 2y = 6 and 2x + 4y = 12.
Solucion: D = 1(4) - 2(2) = 0. The second equation is exactly twice the first, so the equations are dependent.
Respuesta: Infinitely many solutions. Any point on the line x + 2y = 6 is a solution.
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