System of Equations Solver
Solve two linear equations with two unknowns using Cramer's rule.
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Request a ToolHow to Use the System of Equations Solver
Enter the coefficients for two linear equations in the form a*x + b*y = c, and the solution appears instantly.
- Enter equation 1 coefficients. Fill in a1, b1, and c1 for the first equation.
- Enter equation 2 coefficients. Fill in a2, b2, and c2 for the second equation.
- Read the solution. The values of x and y are displayed along with the determinant. If the system has no solution (parallel lines) or infinite solutions (same line), that is indicated.
About Solving Systems of Equations
A system of two linear equations with two unknowns can have exactly one solution (the lines intersect), no solution (the lines are parallel), or infinitely many solutions (the lines are identical). This calculator uses Cramer's rule, which relies on determinants to find the solution.
Cramer's rule states that for the system a1*x + b1*y = c1 and a2*x + b2*y = c2, the solution is x = det_x / det and y = det_y / det, where det = a1*b2 - a2*b1. If the determinant is zero, the system is either inconsistent or dependent.
Frequently Asked Questions
What does "no solution" mean?
If the system has no solution, the two equations represent parallel lines that never intersect. This happens when the determinant is zero but the equations are not proportional.
What does "infinite solutions" mean?
Infinite solutions occur when both equations represent the same line. Every point on that line satisfies both equations. This happens when all three ratios a1/a2, b1/b2, and c1/c2 are equal.
What is Cramer's rule?
Cramer's rule solves a system of linear equations using determinants. For a 2x2 system, x = (c1*b2 - c2*b1) / (a1*b2 - a2*b1) and y = (a1*c2 - a2*c1) / (a1*b2 - a2*b1). It works when the main determinant is non-zero.