System of Equations Calculator (2×2, Free)

Use this free system of equations calculator to solve a 2×2 system of linear equations. Enter the coefficients and you’ll get the solution (x, y), the determinant, both lines in slope-intercept form, a graph of the two lines, and the full elimination steps.

How to use the calculator

1. Enter the coefficients for both equations in the form ax + by = c.
2. Press Solve.
3. See the solution point, the graph, and the step-by-step working.

One solution, none, or infinitely many

If the two lines cross once, there is a single solution. If they are parallel, there is no solution. If they are the same line, there are infinitely many solutions. The calculator detects all three cases using the determinant.

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Frequently asked questions

How does the calculator solve the system?

It uses the determinant D = a1·b2 − a2·b1 and Cramer’s rule for the unique-solution case, and checks consistency when D = 0.

What does ‘no solution’ mean?

The two lines are parallel and never intersect, so no pair (x, y) satisfies both equations.

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Can it show the graph?

Yes — both lines are plotted on one coordinate plane and the intersection point is marked when it exists.

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Read the full lesson: learn the method step by step.

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