System of Equations Calculator
Two equations, two unknowns — and the two ways it fails.
Two equations, two unknowns — and the two ways it fails. Enter six coefficients in the form ax + by = c for both equations.
In the form ax + by = c.
Solution
x = 1, y = 2
One solution — the point where the two lines cross
Solved by Cramer's rule: each unknown is a ratio of two determinants. A non-zero determinant means the two lines cross at exactly one point. A zero determinant means they are parallel, and then there are two genuinely different outcomes. If the equations describe the same line, every point on it is a solution — infinitely many. If they describe distinct parallel lines, there is none at all. Reporting "no solution" for both would be wrong half the time, so this distinguishes them.
How the System of Equations Calculator works
Enter six coefficients in the form ax + by = c for both equations. Where the determinant is zero the page distinguishes parallel lines with no solution from identical lines with infinitely many, because those are genuinely different answers.
Also known as: simultaneous equations calculator · solve for x and y calculator · cramer's rule calculator · two equations two unknowns
Where to go next
The System of Equations question rarely arrives on its own. These are the ones that usually come with it:
- Matrix Determinant Calculator — Any square size, by elimination rather than cofactors.
- Quadratic Equation Calculator — Roots, discriminant and vertex — computed stably.
- Slope Calculator — Slope, intercept and the line's equation from two points.
- Percentage Calculator — Every common percentage question in one place.
Frequently asked questions
How do I solve two simultaneous equations?
By substitution, elimination, or Cramer's rule — each unknown as a ratio of two determinants, which is what this page uses. All three give the same answer when one exists.
What does a zero determinant mean?
The two lines are parallel. If they are the same line every point on it solves the system, giving infinitely many solutions. If they are distinct there is none at all. Reporting 'no solution' for both would be wrong half the time.
What is Cramer's rule?
Replace the coefficient column for the unknown you want with the constants, take that determinant, and divide by the determinant of the original coefficients. It generalises to any size, though elimination is faster beyond 3×3.
Why does my system have no solution?
Because the two equations describe parallel lines that never meet — the same slope with different intercepts. Geometrically there is no crossing point, and algebraically eliminating a variable leaves a contradiction like 0 = 3.
Put this calculator on your own site
Free to use, on any site, commercial or not. Paste this where you want it to appear. It is a plain iframe, so it works in WordPress, Squarespace, Wix, Webflow, Ghost and anything else that accepts HTML.
<iframe src="https://www.thecalclibrary.com/embed/system-of-equations-calculator" width="100%" height="640" style="border:1px solid #e2e8f0;border-radius:12px" loading="lazy" title="System of Equations Calculator"></iframe>
<p style="font:13px/1.5 system-ui,sans-serif;margin:6px 0 0;color:#64748b">Powered by <a href="https://www.thecalclibrary.com/system-of-equations-calculator" style="color:#64748b">System of Equations Calculator</a> from The Calc Library</p>The only condition is that the credit line below the frame stays in place. That one line is what pays for the tool being free — it is how anyone else finds it.
Related calculators
Matrix Determinant Calculator
Any square size, by elimination rather than cofactors.
OpenQuadratic Equation Calculator
Roots, discriminant and vertex — computed stably.
OpenSlope Calculator
Slope, intercept and the line's equation from two points.
OpenPercentage Calculator
Every common percentage question in one place.
Open