Calculator
Enter each equation as ax + by = c. Fractions such as 3/4 are accepted.
Example data table
| Case | Equation 1 | Equation 2 | Expected result | Fill form |
|---|---|---|---|---|
| Unique solution | 2x + 3y = 13 | 4x - y = 5 | x = 2, y = 3 | |
| Parallel lines | 2x + 4y = 8 | x + 2y = 9 | No solution | |
| Same line | 3x - 6y = 12 | x - 2y = 4 | Infinitely many solutions | |
| Fraction input | 1/2x + y = 5 | 3x - 2y = 4 | x = 4, y = 3 |
Formula used
Standard form: a₁x + b₁y = c₁ and a₂x + b₂y = c₂
Determinant: D = a₁b₂ - a₂b₁
Numerators: Dₓ = c₁b₂ - c₂b₁, and Dᵧ = a₁c₂ - a₂c₁
Unique solution: x = Dₓ / D, and y = Dᵧ / D, when D ≠ 0.
Classification: If D = 0 and both Dₓ and Dᵧ are zero, the system is dependent. If D = 0 and at least one transformed numerator is nonzero, the system is inconsistent.
How to use this calculator
- Rewrite both equations in the form ax + by = c.
- Enter every coefficient, including zeros for missing terms.
- Choose automatic elimination or select the variable to remove.
- Set precision and graph range if needed.
- Press the calculate button and read the result above the form.
- Use the graph, CSV, and PDF buttons for review or sharing.
About This Elimination Calculator
A system of two equations often describes two linked unknowns. Each equation gives one condition. The elimination method removes one variable, then solves the remaining simple equation. This calculator follows that process in a clear way. It accepts positive numbers, negative numbers, decimals, and fractions. It also checks special cases, such as parallel lines and identical lines.
Why Elimination Works
Elimination works because equal quantities stay equal after multiplication, addition, or subtraction. You can multiply one equation to match a coefficient in the other equation. Then you subtract or add equations. One variable disappears. The other variable becomes easy to solve. After that, the first value is substituted into either original equation.
Reading the Results
The result panel shows the determinant, classification, ordered pair, and verification. A nonzero determinant means one exact intersection point. A zero determinant needs more checking. If both transformed numerators are also zero, the equations describe the same line. If one numerator is not zero, the lines are parallel and inconsistent.
Graph and Exports
The graph draws both lines on the same coordinate plane. A unique solution appears as an intersection marker. This visual check helps students see why the algebraic result is correct. The CSV export is useful for records. The PDF export is helpful for worksheets, notes, and class review.
Practical Uses
This tool is useful for homework, tutoring, finance models, mixture problems, ticket problems, and balance equations. It can also help teachers show each algebraic move without hiding steps. Always review the entered coefficients before relying on the answer. A misplaced sign can change the whole system.
Input Tips
Use the standard form ax plus by equals c. Enter zero when a term is missing. Choose automatic elimination when you only need the answer. Choose a specific variable when you want to practice a method. Keep rounding high for exact work.
Learning Value
The displayed steps reveal scale factors, subtraction, substitution, and checking. That makes the calculator more than an answer box. It becomes a guided lesson. Students can compare the determinant rule with the classic elimination layout. Both approaches lead to the same conclusion. Use results with care.
FAQs
1. What is the elimination method?
It is a method that removes one variable by adding or subtracting adjusted equations. After one variable disappears, the remaining equation gives the other variable.
2. What form should I enter?
Enter both equations in standard form, ax plus by equals c. Put zero where a variable term is missing.
3. Can I enter fractions?
Yes. You can enter values such as 1/2, -3/4, or 2.5. The calculator converts them before solving.
4. What does determinant zero mean?
A zero determinant means the system does not have one unique intersection. It may have no solution or infinitely many solutions.
5. Why does the graph matter?
The graph shows the two lines visually. A crossing point confirms a unique solution. Parallel or matching lines explain special cases.
6. What is an inconsistent system?
An inconsistent system has no ordered pair that satisfies both equations. In graph form, the lines are usually parallel.
7. What is a dependent system?
A dependent system has infinitely many solutions. Both equations describe the same line, or one equation is a true identity.
8. Can I save the answer?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable summary with steps.