Simultaneous Equations Solver Calculator
Solve two equations in two unknowns, or three equations in three unknowns, such as the loop equations of a circuit. Gives x, y and z
Try your own numbers
Estimate only. Not engineering or safety advice. Check the result against the applicable standard and the equipment data, and have a competent engineer review it before it is relied on. Terms of use.
Open in the calculator with these values
About this calculator
This page calculates x, y, z and 2 more results from number of unknowns, equation 1 coefficient of x, equation 1 coefficient of y, equation 1 coefficient of z and 9 more inputs. Change any value or its unit and the result updates. The steps below show the working for the default values, so you can check each line against your own method.
Limits of this calculator
Use this for the loop or node equations of a circuit with two or three unknown currents or voltages, for finding two constants from two measured points, and for balancing equations. Write each equation as a x + b y + c z = k with the unknowns in the same order, and enter a zero for a missing term. Always put the answer back into the original equations.
Cross-check: 43 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review.
Verification status: Engineering reference
Engineering reference. Formula and method documented and tested against separately coded reference calculations. Not yet Verified: the evidence listed on this page is still needed.
What this calculator checks
- The arithmetic of the published formula for the values you entered, in the units shown.
What remains unverified
- That the inputs suit your application. Take ratings, resistances and constants from the manufacturer data sheet or project specification.
Verification register entry
| Calculation author | Rahul Sharma (author) |
|---|---|
| Technical review | No independent technical review has been carried out |
| Individual professional approval | Not recorded |
| Formula reference | not recorded |
| Standard and edition | Not applicable: Pure mathematics or statistics; no electrical standard governs the method |
| Standard review state | Not applicable |
| Scope-screening note | not recorded |
| Cross-check test cases | 43 reference vectors (tests/reference_vectors.json) |
| Review date | not recorded |
| Next review date | not recorded |
| Status | Engineering reference |
Needed before this calculator can be marked Verified: reviewer name, qualification and review date; exact formula reference; formula source (textbook or standard); next review date.
Formula
Worked example
| Input | Value |
|---|---|
| Number of unknowns | 2 equations, x and y |
| Equation 1: coefficient of x | 2 |
| Equation 1: coefficient of y | 3 |
| Equation 1: constant on the right | 13 |
| Equation 2: coefficient of x | 1 |
| Equation 2: coefficient of y | -1 |
| Equation 2: constant on the right | -1 |
- Equations: 2x + 3y = 13 ; 1x + -1y = -1
- D = a₁b₂ − a₂b₁ = 2 × -1 − 1 × 3 = -5
- x = 2 y = 3
- Check: the largest difference between a left side and its constant is 0
Assumptions
- The equations are linear: each unknown appears to the first power and is not multiplied by another unknown
- The determinant D must not be zero. If it is, the equations either contradict each other or repeat each other, and there is no single solution
- The answer is found by elimination with row swapping, which keeps its accuracy when a coefficient is small; the determinant is shown so you can check by Cramer's rule
- When D is very small compared with the size of the coefficients the system is close to having no solution and a small change in an input changes the answer a lot; a warning is shown
- The check error is the largest amount by which the left side of any equation differs from its constant when the answer is put back in
Use this for the loop or node equations of a circuit with two or three unknown currents or voltages, for finding two constants from two measured points, and for balancing equations. Write each equation as a x + b y + c z = k with the unknowns in the same order, and enter a zero for a missing term. Always put the answer back into the original equations.
Inputs and results
| Input | Unit | Allowed or shown | Note |
|---|---|---|---|
| Number of unknowns | (choice) | 2 equations, x and y; 3 equations, x, y and z | |
| Equation 1: coefficient of x | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000 | Each equation is written a x + b y + c z = k. Enter a zero for a missing term. |
| Equation 1: coefficient of y | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000 | |
| Equation 1: coefficient of z | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000; may be left blank | Used only for 3 unknowns; leave blank for 2. |
| Equation 1: constant on the right | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000 | |
| Equation 2: coefficient of x | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000 | |
| Equation 2: coefficient of y | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000 | |
| Equation 2: coefficient of z | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000; may be left blank | Used only for 3 unknowns; leave blank for 2. |
| Equation 2: constant on the right | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000 | |
| Equation 3: coefficient of x | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000; may be left blank | Equation 3 is used only for 3 unknowns; leave it blank for 2. |
| Equation 3: coefficient of y | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000; may be left blank | |
| Equation 3: coefficient of z | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000; may be left blank | |
| Equation 3: constant on the right | (number) | -1,000,000,000,000 or more, up to 1,000,000,000,000; may be left blank |
| Result | Unit | Allowed or shown | Note |
|---|---|---|---|
| x | (number) | Shown to 10 decimal places | |
| y | (number) | Shown to 10 decimal places | |
| z (3 unknowns only) | (number) | Shown to 10 decimal places | |
| Determinant of the coefficients | (number) | Shown to 10 decimal places | |
| Largest check error (left side minus right side) | (number) | Shown to 6 decimal places |
How it is calculated
Each input is converted to the base unit of its quantity, the formula above is applied in the order shown in the worked example, and each result is shown in the unit you choose. Results are held at full precision and rounded only for display, except where a step selects a standard size, as the worked example shows. A value outside its allowed range is refused with a message instead of giving a result.
How the result changes with equation 1: coefficient of x
| Equation 1: coefficient of x | x | y | z |
|---|---|---|---|
| 0.2 | 3.125 | 4.125 | — |
| 0.5 | 2.8571428571 | 3.8571428571 | — |
| 1 | 2.5 | 3.5 | — |
| 2 | 2 | 3 | — |
| 4 | 1.4285714286 | 2.4285714286 | — |
| 8 | 0.9090909091 | 1.9090909091 | — |
| 20 | 0.4347826087 | 1.4347826087 | — |
Other inputs are the same as in the worked example above.
Standards and further reading
Complex numbers, rectangular and polar form and angle conversion are standard mathematics, not the subject of a standard. Standards apply when the result feeds an electrical design, and they set the method there.
Frequently asked questions
What do I need to enter?
Enter number of unknowns, equation 1: coefficient of x, equation 1: coefficient of y, equation 1: coefficient of z, equation 1: constant on the right, equation 2: coefficient of x, equation 2: coefficient of y, equation 2: coefficient of z, equation 2: constant on the right, equation 3: coefficient of x, equation 3: coefficient of y, equation 3: coefficient of z and equation 3: constant on the right. Change the unit of a value with its selector and the result stays the same.
What does this calculator assume?
The equations are linear: each unknown appears to the first power and is not multiplied by another unknown. The determinant D must not be zero. If it is, the equations either contradict each other or repeat each other, and there is no single solution. The answer is found by elimination with row swapping, which keeps its accuracy when a coefficient is small; the determinant is shown so you can check by Cramer's rule. When D is very small compared with the size of the coefficients the system is close to having no solution and a small change in an input changes the answer a lot; a warning is shown. The check error is the largest amount by which the left side of any equation differs from its constant when the answer is put back in.
Which calculators should I use next?
Common next steps are Linear Interpolation Calculator, Quadratic Equation Solver, Complex Numbers (Rectangular and Polar) and Resistors in Parallel. The related links at the bottom of this page open them.
How accurate are the results?
Every figure comes from the same tested engine as the Elecmetric calculators and is meant for estimating. 43 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review. Check design values against the standard and the equipment data you work to.
Formula guides and reference
Longer explanations with worked examples and tables, written for the same topic.
- Kirchhoff’s Current and Voltage Laws: the two circuit laws with worked examples
- Thévenin and Norton Formulas: Thévenin and Norton equivalent circuit formulas: open-circuit voltage, equivalent resistance
- Quadratic, Logarithm and Power Formulas: Formulas for the quadratic equation, logarithms and exponents with electrical examples: real and complex roots, decibels
More Engineering Mathematics calculatorsAll engineering calculatorsSimultaneous Equations Solver in Elecmetric