Formula
ax + b = c ⇒ x = (c − b) ÷ aD = a₁b₂ − a₂b₁; x = (c₁b₂ − c₂b₁) ÷ D; y = (a₁c₂ − a₂c₁) ÷ D
- For one equation, a ≠ 0
- For two equations, D = 0 means no single solution (parallel lines or the same line)
Move everything but x to one side and divide; for two unknowns, Cramer's rule gives x and y from determinants.
Worked examples
2x + 3 = 11
- Type of problem
- One equation: ax + b = c
- a
- 2
- b
- 3
- c
- 11
x
4
Check: 2 × 4 + 3 = 11
2x + y = 8 and x − y = 1
- Type of problem
- Two equations: a₁x + b₁y = c₁ and a₂x + b₂y = c₂
- a₁
- 2
- b₁
- 1
- c₁
- 8
- a₂
- 1
- b₂
- −1
- c₂
- 1
Solution
x = 3, y = 2
Determinant: −3
Check, equation 1: 2×3 + 1×2 = 8
Check, equation 2: 1×3 + −1×2 = 1
How it works
A linear equation in one unknown, ax + b = c, is solved by subtracting b from both sides and dividing by a. If a is zero the equation has either no solution or infinitely many, depending on whether b equals c, and the calculator says which.
Two equations in x and y describe two straight lines. They usually cross at exactly one point, which Cramer's rule finds from the determinant D = a₁b₂ − a₂b₁. When D is zero the lines are parallel (no solution) or identical (infinitely many solutions), and the tool reports which case you have. Every answer is checked by substituting it back into your equations.
For equations with an x² term use the Quadratic Equation Solver. For a ratio or percentage word problem the Proportion Calculator may be quicker, and the Scientific Calculator helps evaluate the numbers.
Assumptions and limits
- Coefficients are real numbers.
- Near-zero determinants caused by rounding in your own numbers are treated as exactly zero only when the arithmetic gives zero.