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This is the most approachable entry point on the site for solving a system of linear equations: enter each equation directly in standard form (a·x + b·y [+ c·z] = d) rather than filling in a bare coefficient matrix, and get the full Cramer's-rule solution.
How it works
The equations' coefficients form a matrix, and Cramer's rule solves for each variable by replacing that variable's column with the constants column and dividing the resulting determinant by the main determinant.
- Enter number of equations/variables.
- Enter equation 1: x coefficient.
- Enter equation 1: y coefficient.
- Enter equation 1: z coefficient (only if 3 equations).
- Enter equation 1: right side.
- Enter equation 2: x coefficient.
- Enter equation 2: y coefficient.
- Enter equation 2: z coefficient (only if 3 equations).
- Enter equation 2: right side.
- Enter equation 3: x coefficient (only if 3 equations).
- Enter equation 3: y coefficient (only if 3 equations).
- Enter equation 3: z coefficient (only if 3 equations).
- Enter equation 3: right side (only if 3 equations).
- Click Calculate to see your results.
Examples
A 2-equation system
2x+3y=7 and x-y=1 solves to x=2, y=1.
A 3-equation system
x+y+z=6, 2y+5z=-4, and 2x+5y-z=27 solves to x=5, y=3, z=-2.
Who should use it
- Algebra coursework on solving systems of linear equations.
- A quick, approachable way to solve a system copied directly from a word problem.
Industry applications
- Mathematics education
Advantages
- More approachable, equation-based framing than the matrix-grid tools.
- Shows every Cramer's rule determinant explicitly.
Limitations
- Limited to 2 or 3 equations/variables — for larger systems, use the matrix-based Gaussian Elimination or Gauss-Jordan tools.
Common mistakes to avoid
- Entering an equation not in standard form (e.g. "y=2x+3" instead of "-2x+y=3"), which would need rearranging first.
Best practices
- Rearrange every equation into standard form (variables on the left, constant on the right) before entering the coefficients.
Tips
- If you already have your system as a bare coefficient matrix, the site's dedicated Cramer's Rule or Gaussian Elimination calculators may be a more direct fit.