System of Equations Calculator
Solve systems of 2 or 3 linear equations in x, y and z. Type the equations as you would write them, and get the solution, the determinant and the elimination steps.
Results
| Item | Value |
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Steps
| Step | Working |
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Quick Answer
To solve a system of linear equations, eliminate one variable at a time. For 2x + 3y = 8 and x − y = −1, the solution is x = 1 and y = 2. Check: 2(1) + 3(2) = 8 and 1 − 2 = −1.
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Table of Contents
- System of Equations Calculator
- Quick Answer
- Related Calculators
- How This System of Equations Calculator Works
- How to Use the System of Equations Calculator
- Methods
- Key Terms
- Worked Example
- Real-Life Examples
- Elimination vs Substitution
- Sample Systems
- Tips for Entering Equations
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This System of Equations Calculator Works
Type 2 or 3 linear equations, one per line, using x, y and z. The calculator reads each equation, moves every term to the left side, and builds a matrix. It then uses Gauss–Jordan elimination to clear each variable from the other rows, shows each step, and gives the solution as exact fractions where possible.
Quick explanation: a solution makes every equation true at the same time.
- One solution: lines or planes meet at a single point.
- No solution: parallel lines or planes never meet.
- Infinitely many: the equations describe the same line or plane.
How to Use the System of Equations Calculator
- Type your equations, one per line.
- Use x and y for 2 equations, or x, y and z for 3.
- Press Calculate.
- Read the solution and the determinant.
- Follow the elimination steps to check your work.
Methods
- Elimination: add multiples of one equation to another to cancel a variable.
- Substitution: solve one equation for a variable, then put it into the others.
- Cramer’s rule (2 × 2): x = (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁), y = (a₁c₂ − a₂c₁) ÷ (a₁b₂ − a₂b₁)
- Determinant: if it is 0, there is no single solution.
Key Terms
- Linear equation: an equation where variables appear only to the first power.
- System of equations: two or more equations that share the same variables.
- Determinant: a number from the coefficients that shows if there is a unique solution.
- Pivot: the leading coefficient used to eliminate a variable from other rows.
- Consistent system: a system with at least one solution.
Worked Example
Solve x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.
- Eliminate x from rows 2 and 3 using row 1
- Eliminate y, then z, working down and back up
- Solution: x = 1, y = 2, z = 3
Real-Life Examples
- Ticket sales: adult and child tickets: a + c = 100 and 12a + 7c = 1,000 gives a = 60 and c = 40.
- Mixtures: mixing two solutions to reach a target strength.
- Break-even: where two cost lines cross, such as two phone plans.
- Homework: check answers to elimination or substitution problems.
Elimination vs Substitution
- Elimination: best when coefficients line up or are easy to match.
- Substitution: best when one variable already has a coefficient of 1.
- Matrices: the same elimination, written in rows. Try our Matrix Calculator.
Sample Systems
| Equations | Solution |
|---|---|
| x + y = 10, x − y = 2 | x = 6, y = 4 |
| 2x + 3y = 8, x − y = −1 | x = 1, y = 2 |
| x + 2y = 4, 2x + 4y = 8 | Infinitely many |
| x + y = 1, x + y = 3 | No solution |
| x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 | x = 1, y = 2, z = 3 |
Tips for Entering Equations
- One equation per line: each needs exactly one = sign.
- Variables: use x and y for 2 equations, and x, y and z for 3.
- Terms on both sides: fine. 3x = 9 − y works.
- Decimals: write 0.5x rather than 1/2x.
Common Mistakes to Avoid
- Sign errors. Watch minus signs when moving terms across the = sign.
- Multiplying only one side. Multiply every term in the equation.
- Stopping early. Substitute back to find every variable.
- Not checking. Put the answers into each original equation.
- Assuming every system has one answer. Some have none or infinitely many.
Frequently Asked Questions
Use elimination or substitution to remove one variable, solve for the other, then substitute back.
Eliminate one variable to get two equations in two variables, solve those, then substitute back.
The lines or planes are parallel and never meet.
The equations describe the same line or plane, so every point on it works.
If the determinant of the coefficients is 0, there is no single solution.
Yes. The calculator moves everything to one side for you.
Use x, y and z. Rename other variables before entering.
No. It solves linear equations only, with no squares or products of variables.
A formula that solves a system using determinants. It works when the determinant is not 0.
Fractions are exact. A decimal is shown too when the answer is not a whole number.
A method that uses row operations to turn the system into one where each variable can be read directly.
Put the values back into every original equation and make sure both sides match.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, System of linear equations | Solving methods and types of solutions. | https://en.wikipedia.org/wiki/System_of_linear_equations |
| Wikipedia, Gaussian elimination | Row reduction step by step. | https://en.wikipedia.org/wiki/Gaussian_elimination |