System of Equations Solver
Solve a 2×2 linear system: a1x + b1y = c1 and a2x + b2y = c2.
What is a system of equations solver?
A system of equations solver finds the x and y values that satisfy two linear equations at the same time — the point where the two lines cross.
How to use the system of equations solver
Enter the a, b and c values of each equation (a1x + b1y = c1 and a2x + b2y = c2) and click Solve. The solver shows x, y and the determinant.
System of equations formula
Determinant D = a1b2 − a2b1. Then x = (c1b2 − c2b1) ÷ D and y = (a1c2 − a2c1) ÷ D. If D = 0 there is no unique solution: the lines are parallel or the same line.
Example
x + y = 5 and x − y = 1: the determinant is 1 × (−1) − 1 × 1 = −2, so x = 3 and y = 2. Check: 3 + 2 = 5 and 3 − 2 = 1.
Frequently Asked Questions
What is the solution of x + y = 5 and x − y = 1?
The solution is x = 3 and y = 2. Try it on the solver above.
What does a determinant of 0 mean?
A determinant of 0 means there is no unique solution: the lines are parallel (no solution) or the same line (infinitely many solutions).
How do you check the solution?
Put x and y back into both equations. For x = 3, y = 2: 3 + 2 = 5 and 3 − 2 = 1, so both equations are satisfied.
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