System of Equations Solver
Solve two or three simultaneous linear equations by elimination, substitution, cross-multiplication, Cramer's rule or the matrix inverse, with every step checked.
About the System of Equations Solver
Type the equations as you'd write them (2x + 3y = 13) or as their numbers (2, 3, 13). Pick the method your textbook uses: every method gives the same answer, and each line of working is proved before it's shown.
A pair of linear equations has exactly one solution when a₁/a₂ ≠ b₁/b₂ (the lines cross), none when a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines), and infinitely many when all three ratios are equal (the same line).
The formula
x = (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁), y = (a₁c₂ − a₂c₁) ÷ (a₁b₂ − a₂b₁)
- a₁x + b₁y = c₁
- first equation
- a₂x + b₂y = c₂
- second equation
- a₁b₂ − a₂b₁
- the determinant D
Worked example
2x + 3y = 13 and x − y = −1
- Multiply the second by 3: 3x − 3y = −3
- Add: 5x = 10, so x = 2
- Then 2(2) + 3y = 13, so y = 3
Questions people ask
Which method should I use?
Elimination is quickest by hand; substitution is easiest when one variable has coefficient 1; Cramer's rule and the matrix inverse suit small systems and are a good check.
Can I use fractions?
Yes: 1/2x + y = 3 or 0.5, 1, 3. Answers stay exact.