Solve 2×2 and 3×3 systems of linear equations using Cramer's Rule. Compute the coefficient determinant and the solution values instantly.
| System | Solution |
|---|---|
| 2x + y = 5, x - y = 1 | x = 2, y = 1 |
| 3x + 2y = 16, x - y = 2 | x = 4, y = 2 |
| x + y = 7, 2x - y = 5 | x = 4, y = 3 |
Cramer's Rule is a method for solving a square system of linear equations using determinants. It works when the coefficient matrix is square and its determinant is not zero.
Instead of using elimination, Cramer's Rule computes one determinant for the coefficient matrix and additional determinants for matrices where one column at a time is replaced by the constants column.
This makes it especially useful for small systems like 2×2 and 3×3 problems, where determinant formulas are manageable and easy to verify.
Write the coefficients of the variables into a square matrix A and place the constants on the right-hand side.
Find the determinant of the coefficient matrix. If det(A) = 0, Cramer's Rule does not give a unique solution.
To find x, replace the x-column with the constants column and compute det(X). Repeat for y and z if needed.
Use x = det(X) ÷ det(A), y = det(Y) ÷ det(A), and z = det(Z) ÷ det(A).
Solving a 2×2 system
You can use it for square systems where the coefficient determinant is not zero.
Then the system does not have a unique solution by Cramer’s Rule. It may have no solution or infinitely many solutions.
It is usually best for small systems such as 2×2 and 3×3 because determinant calculations become expensive for larger matrices.
Determinants measure whether the coefficient matrix is invertible and allow each variable to be isolated through replacement determinants.
Yes. It accepts valid decimal values, zero, and negative numbers.
det(A) is the determinant of the coefficient matrix. det(X) is the determinant after replacing the x-column with the constants column.
Related Tools
Solve absolute value equations.
Solve absolute value inequalities.
Apply associative property.
Multiply using box method.
Complete the square method.
Find complex roots.