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Cramer's Rule Calculator.

Solve a 2×2 linear system using Cramer's rule: ax + by = e and cx + dy = f.

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Set your values

Results update as you type.

Determinant: 5

Determinant

5.000000
x: 1.8

x

1.800000
y: 1.4

y

1.400000

Results update automatically as you type.

Use Cases

Solve homework problems

Quickly check your work when solving 2x2 linear systems in algebra or linear algebra courses.

Example: Solve 2x + 3y = 8 and 4x - y = 1.

Find intersection of two lines

Given two linear equations, find the point where the lines intersect, which is the solution to the system.

Example: Find the intersection of y = 2x + 1 and y = -x + 4.

Frequently Asked Questions

What is Cramer's rule?
Cramer's rule is a formula for solving a system of linear equations using determinants. For a 2x2 system ax+by=e and cx+dy=f, it computes x and y as ratios of determinants: x = (ed - bf)/(ad - bc) and y = (af - ec)/(ad - bc).
When does Cramer's rule fail?
Cramer's rule fails when the determinant of the coefficient matrix (ad - bc) equals zero. In that case, the system either has no solution or infinitely many solutions, and the calculator cannot provide a unique solution.
Can I use this calculator for systems with more than two variables?
No, this calculator is specifically designed for 2x2 systems (two equations and two unknowns). For larger systems, you would need a more general solver.

Tips & Common Mistakes

Tips

  • Ensure you enter the coefficients correctly: a, b, c, d are the coefficients of x and y in the two equations, and e, f are the constants on the right side.
  • If the determinant (ad - bc) is zero, the system may have no solution or infinite solutions. Check your equations for consistency.
  • Use the calculator to verify your manual calculations, but also practice solving by substitution or elimination to understand the concepts.

Common Mistakes to Avoid

  • Mixing up the order of coefficients: make sure a and b are from the first equation, c and d from the second, and e and f are the constants.
  • Forgetting that the determinant must be non-zero for a unique solution; if it's zero, the calculator will not give a valid answer.
  • Entering negative numbers incorrectly: use the minus sign (-) before the number, not a hyphen or parentheses.

Last updated: August 13, 2026