Calculator Input
Example Data Table
This example gives the solution x = 2, y = -1, and z = 3.
| Equation | x coefficient | y coefficient | z coefficient | Constant |
|---|---|---|---|---|
| 2x + y - z = 0 | 2 | 1 | -1 | 0 |
| x - 3y + 2z = 11 | 1 | -3 | 2 | 11 |
| 3x + 2y + z = 7 | 3 | 2 | 1 | 7 |
Formula Used
A 3x3 system is written as:
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
The coefficient determinant is:
For a unique solution, Cramer’s rule is:
y = det(Ay) / det(A)
z = det(Az) / det(A)
If det(A) equals zero, the calculator compares matrix ranks. If Rank(A) is less than Rank(A|B), there is no solution. If both ranks are equal and less than three, there are infinitely many solutions.
How to Use This Calculator
- Write each equation in the form ax + by + cz = d.
- Enter the x, y, and z coefficients for all three equations.
- Enter each constant from the right side of the equation.
- Select decimal precision and tolerance.
- Keep step output enabled if you want a detailed explanation.
- Press the calculate button.
- Review the solution type, determinant values, ranks, and final values.
- Use the CSV or PDF button to save the result.
Complete Guide to 3x3 Systems
What a 3x3 System Means
A 3x3 system of equations has three unknowns and three linear equations. It often appears in algebra, physics, economics, and engineering. Each equation describes a plane in three dimensional space. The solution is the shared point where the planes meet.
Why This Calculator Helps
This calculator is built for careful solving. It accepts all nine coefficients and three constants. It then checks the determinant and matrix ranks. These checks are important. They tell you whether the system has one solution, no solution, or many solutions.
Unique Solutions
A unique solution happens when the coefficient determinant is not zero. In that case, the planes meet at one point. The calculator can use Cramer’s rule to find x, y, and z. It also reports the determinant values used in the calculation.
Special Cases
Some systems do not have a single answer. Parallel or conflicting planes may create no shared point. Dependent equations may create a line or plane of solutions. Rank comparison helps identify these cases. If the coefficient rank differs from the augmented rank, the system is inconsistent. If both ranks are equal but lower than three, the system is dependent.
Export and Review
The tool also supports practical review. You can choose decimal precision. You can enable step output. You can export results for records. The CSV file is useful for spreadsheets. The PDF file is useful for printable homework notes.
Input Tips
Use the example table before entering your own data. It shows how coefficients map into the form ax + by + cz = d. Keep signs accurate. Use negative values when a term is subtracted. Use zero when a variable is missing from an equation.
Learning Value
This calculator is not only for final answers. It helps learners understand structure. Determinants reveal whether division is safe. Ranks reveal whether equations agree. The step table makes the process easier to check. With these details, you can verify manual work and build stronger confidence in linear algebra.
Classroom Use
The page is also helpful for teachers. It creates consistent worked examples. Students can compare their row work against the reported classifications. Small rounding choices can change displayed decimals. The tolerance option helps control that issue. For exact classroom proofs, keep fractional manual notes beside the calculator output when needed.
FAQs
1. What is a 3x3 system of equations?
It is a group of three linear equations with three unknown variables. The variables are usually x, y, and z. The goal is to find values that satisfy all three equations at the same time.
2. When does a 3x3 system have one solution?
It has one solution when the determinant of the coefficient matrix is not zero. This means the three planes meet at one exact point.
3. What does det(A) mean?
det(A) is the determinant of the coefficient matrix. It shows whether the system can be solved uniquely using determinant based methods like Cramer’s rule.
4. What happens if det(A) is zero?
If det(A) is zero, the system may have no solution or infinitely many solutions. The calculator then compares coefficient and augmented ranks.
5. What is an inconsistent system?
An inconsistent system has no shared solution. In rank terms, this happens when the coefficient matrix rank is less than the augmented matrix rank.
6. What is a dependent system?
A dependent system has infinitely many solutions. Its equations overlap in a way that does not create one single intersection point.
7. Can I enter decimal coefficients?
Yes. You can enter whole numbers, decimals, negative values, and zero. Use zero when a variable is missing from an equation.
8. Why use CSV and PDF exports?
CSV exports help with spreadsheets and data records. PDF exports are better for printing, sharing, and saving a formatted solution report.