Solve simultaneous 2x2 and 3x3 systems of linear equations using Cramer’s Rule and matrix determinants with step-by-step work.
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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.
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.
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
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).
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).
Yes, switch to the 3×3 tab to solve three simultaneous equations with variables x, y, and z.
Substitute the values of x and y into each equation and check if both sides evaluate to identical numbers.
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