Read the result
A perfect exact solution has residual 0. With decimals, small residuals are usually acceptable, but scale matters.
Solution checks
Linear System Residual Checker
Test a proposed solution by calculating Ax - b and measuring the leftover error.
Formula
r = Ax - b; ||r|| = sqrt(r1^2 + r2^2 + r3^2)
Result
Worked substitution
A residual close to zero means the proposed solution fits the original equations well.
Where it helps
Use it to check rounded answers from elimination, iteration, or calculator output.
Common slip
A residual of 0.001 may be tiny for large engineering units but large for a system with coefficients near 0.01.
Try it
Change x3 from -1 to -0.99 in the default system and see which equation notices most.