How to use the System of Equations Calculator (2 Variables)
- Write each equation in the form ax + by = c.
- Enter a₁, b₁ and c₁ for the first equation.
- Enter a₂, b₂ and c₂ for the second equation.
- Read x, y and the determinant, or the message about no or many solutions.
Formula used
With the determinant D = a₁b₂ − a₂b₁: x = (c₁b₂ − c₂b₁) ÷ D and y = (a₁c₂ − a₂c₁) ÷ D (Cramer’s rule). If D = 0 the lines are parallel or the same line.
Worked example
Here is the System of Equations Calculator (2 Variables) worked through with the example values shown above, so you can check each step by hand.
| Inputs | |
|---|---|
| Equation 1: coefficient of x (a₁) | 2 |
| Equation 1: coefficient of y (b₁) | 3 |
| Equation 1: constant (c₁) | 13 |
| Equation 2: coefficient of x (a₂) | 1 |
| Equation 2: coefficient of y (b₂) | -1 |
| Equation 2: constant (c₂) | -1 |
| Result | |
| Solution | x = 2, y = 3 |
| x | 2 |
| y | 3 |
| Determinant (a₁b₂ − a₂b₁) | -5 |
A worked example
The default equations are 2x + 3y = 13 and x − y = −1. The determinant is 2 × (−1) − 1 × 3 = -5. Then x = (13 × (−1) − (−1) × 3) ÷ (−5) = 2 and y = (2 × (−1) − 1 × 13) ÷ (−5) = 3. The calculator gives x = 2, y = 3.
Check it: 2(2) + 3(3) = 13 and 2 − 3 = −1. Both are true, so the pair satisfies both equations.
The method: Cramer’s rule
The calculator uses Cramer’s rule. It first computes the determinant D = a₁b₂ − a₂b₁. If D is not zero there is exactly one solution, x = (c₁b₂ − c₂b₁) ÷ D and y = (a₁c₂ − a₂c₁) ÷ D.
You can reach the same answer by substitution or elimination. Cramer’s rule is simply a fast formula for the elimination steps, which is why it suits a calculator.
No solution and infinitely many solutions
When the determinant is zero the two lines have the same slope. If the equations describe the same line, every point on it works. For x + 2y = 3 and 2x + 4y = 6, the calculator reports: Infinitely many solutions (same line). If the lines are parallel but different, as in x + 2y = 3 and 2x + 4y = 7, it reports: No solution (parallel lines).
In real problems, no solution often means the conditions contradict each other, while infinitely many means the second equation adds no new information.
Using it for word problems
Many word problems reduce to two equations. For example, if 3 notebooks and 2 pens cost 12 and one notebook costs 1 more than a pen, the equations are 3x + 2y = 12 and x − y = 1, giving x = 2.8, y = 1.8. Notebooks cost 2.8 and pens 1.8.
Define x and y clearly, write one equation for each fact, and put them in the form ax + by = c before entering them. Move terms across if needed, and remember to change signs when you do.
Frequently asked questions
What if an equation has no y term?
Enter 0 as its y coefficient.
Can it solve three equations?
No, this calculator handles two equations with two unknowns.
What does a determinant of zero mean?
The lines are parallel or identical, so there is no unique solution.
Does it accept decimals and negatives?
Yes, enter them as normal numbers.
How do I check my answer?
Substitute x and y into both equations. Both must be true.
Why does it show a rounded value?
Results are shown to ten significant digits, so long decimals are trimmed.
This calculator gives estimates for information only. Actual figures from a bank, employer, tax authority or doctor may differ. It is not financial, tax, legal or medical advice.