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- Solves 2 equations with 2 variables (x, y) or 3 equations with 3 variables (x, y, z).
- Uses Cramer’s rule and shows every determinant.
- Gives exact fraction answers for whole-number coefficients.
- Tells you when there is no solution or infinitely many.
How to use the system of equations calculator
Choose 2 or 3 variables
For x and y, or x, y and z.
Write each equation in standard form
Move terms so it looks like ax + by = c (or ax + by + cz = d).
Enter the coefficients
Type the number in front of each variable and the constant after the equals sign. Use 0 for a missing variable and −1 for a minus sign alone.
Read the solution
See the values with each step of Cramer’s rule.
How Cramer’s rule works
Cramer’s rule solves a linear system with determinants. First find D, the determinant of the coefficients. Then replace one column at a time with the constants to get Dx, Dy (and Dz). Each variable is its determinant divided by D. If D = 0, there is no single solution.
Cramer’s rule for 2 equations
For ax + by = e and cx + dy = f:
D = ad − bc
x = (ed − bf) ÷ D, y = (af − ec) ÷ D
Example 1: 2x + 3y = 13 and x − y = −1
Two variables.
- D 2 × (−1) − 3 × 1 = −5
- Dx −10, so x = 2
- Dy −15, so y = 3
Example 2: three variables
x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2
- D 7
- Solution x = 1, y = 2, z = 3
Example 3: fraction answer
3x + 2y = 1 and x − y = 4
- Solution x = 9/5, y = −11/5
Where systems of equations are used
Algebra homework
Check answers to elimination and substitution problems.
Mixture problems
Find amounts of two or three ingredients.
Business
Find break-even points and pricing mixes.
Physics and circuits
Solve for unknown forces or currents.
Graph a line with the slope calculator, solve one equation with the quadratic formula calculator, or work with fractions in the fraction calculator.
System of equations calculator FAQ
How do you solve a system of two equations?
Use substitution, elimination or Cramer’s rule. All three give the same answer when there is one solution.
What does it mean if there is no solution?
The lines (or planes) never meet, for example two parallel lines like x + 2y = 3 and 2x + 4y = 7.
What does infinitely many solutions mean?
The equations describe the same line or plane, so every point on it works.
What is a determinant?
A number calculated from the coefficients. If it is 0, the system has no unique solution.
Can I use decimals?
Yes. Answers are then shown as decimals instead of fractions.
How do I enter x − y = −1?
Enter 1 for x, −1 for y and −1 for the constant.