Simultaneous equations solver (2 unknowns)
Enter the coefficients of two linear equations in the form ax + by = c and this solver returns x and y with the elimination working shown. It also spots the awkward cases exam questions like to include — parallel lines with no solution and identical lines with infinitely many solutions.
Enter both equations as ax + by = c
Equation 1
Equation 2
x = 3, y = 2
Determinant: a₁b₂ − a₂b₁ = -5 (non-zero, so the lines cross at exactly one point).
Check: substituting back gives 12 = 12 and 1 = 1.
The method behind it
- Determinant: D = a₁b₂ − a₂b₁. If D ≠ 0 there is exactly one solution.
- x = (c₁b₂ − c₂b₁) / D and y = (a₁c₂ − a₂c₁) / D.
- D = 0 means the lines are parallel (no solution) or identical (infinitely many).
How to use this tool
1. Write both equations in ax + by = c form
Rearrange each equation so the x and y terms are on the left and the number is on the right.
2. Enter the six coefficients
Type a, b and c for each equation. Fractions can be entered as decimals.
3. Check the solution
Substitute the values back into both original equations — the solver shows this check so you can copy the verification step into your working.
Frequently asked questions
How do you solve simultaneous equations by elimination?
Scale one or both equations so the coefficients of x (or y) match, then add or subtract the equations to eliminate that variable. Solve the resulting single-variable equation and substitute back to find the other unknown.
When do simultaneous equations have no solution?
When the two lines are parallel — the coefficients of x and y are in the same ratio but the constants are not. Algebraically the determinant a₁b₂ − a₂b₁ equals zero while the equations are inconsistent.
What does infinitely many solutions mean?
Both equations describe the same line, so every point on that line satisfies them. This happens when one equation is a multiple of the other.
Should I use elimination or substitution?
Elimination is usually faster when both equations are already in ax + by = c form. Substitution is better when one equation is already written as y = … or x = …, which is common in linear-and-quadratic pairs.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.