These sample systems show typical inputs and expected outputs.
| Mode | System (compact) | Solution | Notes |
|---|---|---|---|
| 2 vars | 2x + 3y = 13; x - y = 1 | x = 4, y = 1 | Isolate x from eq2, then substitute. |
| 2 vars | x + 2y = 5; 2x + 4y = 10 | Infinitely many | Second equation is a multiple of the first. |
| 3 vars | x + y + z = 6; 2x + y - z = 3; x - y + 2z = 9 | x = 1, y = 2, z = 3 | Reduce to two variables, then back-substitute. |
For a 2-variable system: a1x + b1y = c1 and a2x + b2y = c2. Isolate one variable from equation (1), substitute into equation (2), solve the single-variable equation, then back-substitute to find the other variable.
- If isolating x: x = (c1 - b1y)/a1
- If isolating y: y = (c1 - a1x)/b1
- For 3 variables, isolate one variable in equation (1) and substitute into equations (2) and (3).
- Select 2-variable or 3-variable mode.
- Enter coefficients and constants using numbers or fractions.
- Optionally choose which variable to isolate first.
- Press Submit to see the solution and step list.
- Use Download CSV or Download PDF to export results.
1) What is the substitution method?
It solves linear systems by isolating one variable in one equation and replacing it in the other equation(s). This reduces variables until you can solve directly, then you back-substitute to find remaining values.
2) Can I enter fractions like 3/5?
Yes. Use integers, decimals, or fractions such as 3/5 or -7/4. The calculator keeps exact fraction arithmetic and also shows decimals for quick reading.
3) What does “no solution” mean?
No solution means the equations contradict each other, like parallel lines. After substitution, you may get a false statement such as 0 = 5, showing the system is inconsistent.
4) What does “infinitely many solutions” mean?
It means the equations describe the same relationship. After substitution, you may get 0 = 0. In that case, many (x, y) pairs satisfy both equations.
5) Why does the calculator choose an “auto” isolation?
Auto tries to pick a stable path: it avoids dividing by zero and prefers isolating a variable whose coefficient makes substitution easier. You can override it using the preference dropdown.
6) How does it handle 3 variables?
It isolates one variable from the first equation, substitutes into the other two equations, and forms a 2-variable system. Then it solves that system and back-substitutes to recover the third variable.
7) Are decimals rounded?
Fractions are exact. Decimals are displayed with fixed precision for readability, so small rounding differences can appear in the decimal view only, not in the fraction result.
8) What should I do if an input is blank?
Blank fields are treated as zero. If that makes an equation invalid, you may see an input error. Fill required coefficients for the variables you want to model.