Quick answer
A two-variable linear system a₁x + b₁y = c₁, a₂x + b₂y = c₂ can be solved with Cramer's rule. x + y = 5 and x − y = 1 give x = 3, y = 2.
Enter the coefficients of both equations. The calculator returns the solution.
Related tools: Slope Calculator: Slope, angle, distance and line equation through two points. LCM Calculator: Least common multiple of two or more numbers. Factor Calculator: List every factor of a positive whole number.
How to use the System of Equations Calculator
- Enter a₁.
- Enter b₁.
- Enter c₁.
- Enter a₂.
- Enter b₂.
- Enter c₂.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| a₁ | A number | — |
| b₁ | A number | — |
| c₁ | A number | — |
| a₂ | A number | — |
| b₂ | A number | — |
| c₂ | A number | — |
Formula and method
Formula
a₁x + b₁y = c₁; a₂x + b₂y = c₂ → Cramer's rule
IEEE-754 double precision; shown to at most 6 decimal places.
x = (c₁b₂ − c₂b₁) ÷ D, y = (a₁c₂ − a₂c₁) ÷ D, D = a₁b₂ − a₂b₁. D = 0 means no unique solution.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: a₁ = 1; b₁ = 1; c₁ = 5; a₂ = 1; b₂ = -1; c₂ = 1.
| Result | Value |
|---|---|
| Solution | x = 3, y = 2 |
Example 2
Inputs: a₁ = 2; b₁ = 3; c₁ = 13; a₂ = 1; b₂ = -1; c₂ = 1.
| Result | Value |
|---|---|
| Solution | x = 3.2, y = 2.2 |
The first line is the headline result (solution). IEEE-754 double precision; shown to at most 6 decimal places.
Cramer's rule
D = a₁b₂ − a₂b₁ = (1)(−1) − (1)(1) = −2. x = (c₁b₂ − c₂b₁) ÷ D = (−5 − 1) ÷ −2 = 3 and y = (a₁c₂ − a₂c₁) ÷ D = (1 − 5) ÷ −2 = 2.
No unique solution
If D = 0 the lines are parallel (no solution) or identical (infinitely many), and the calculator reports that there is no unique solution. Check by substituting: 3 + 2 = 5 and 3 − 2 = 1.
How the result changes with a₁
Holding every other input at the first worked example, this table changes only a₁. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| a₁ | Solution |
|---|---|
| 0.5 | x = 4, y = 3 |
| 1 | x = 3, y = 2 |
| 2 | x = 2, y = 1 |
| 4 | x = 1.2, y = 0.2 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- a₁: entering "1" shows "The equations have no unique solution (the lines are parallel or identical)." beside that field and no result.
Choosing between the System of Equations Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
| [Quadratic Formula Calculator](/math/quadratic-formula-calculator/) | a, b and c | Roots |
| [Log Calculator](/math/log-calculator/) | number x and base b | log base 2 of 8 |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Parallel or identical lines have no unique solution
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for a₁ to confirm this.
Frequently asked questions
What if the lines are parallel?
There is no solution.
Can I use elimination instead?
Yes, it gives the same answer.
How many solutions can there be?
None, one or infinitely many.
What does the System of Equations Calculator calculate?
Solve two linear equations in two unknowns. It returns Solution.
What formula does the System of Equations Calculator use?
a₁x + b₁y = c₁; a₂x + b₂y = c₂ → Cramer's rule. x = (c₁b₂ − c₂b₁) ÷ D, y = (a₁c₂ − a₂c₁) ÷ D, D = a₁b₂ − a₂b₁.
What do I need to enter in the system of Equations Calculator?
You enter a₁, b₁, c₁, a₂, b₂ and c₂.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Parallel or identical lines have no unique solution.
What is an example of a system of Equations result?
With a₁ 1, b₁ 1, c₁ 5, a₂ 1 and b₂ -1, the solution is x = 3, y = 2.
Related calculators
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
- Log Calculator — Logarithm of a number to any base.
- GCF Calculator — Greatest common factor (greatest common divisor) of two or more numbers.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.