System of equations: step-by-step solving
A system of equations is a set of two or more equations sharing variables. Solving it means finding the values of variables that satisfy all equations simultaneously.
Solving methods
- Substitution: isolate one variable in one equation and substitute into the other.
- Elimination (addition/subtraction): multiply equations to cancel one variable.
- Equalization: isolate the same variable in both equations and set them equal.
- Matrix method (Gauss-Jordan): for systems with more variables.
- Cramer's rule: using determinants.
Geometric interpretation
Each linear equation with two variables represents a line. The system has a unique solution if the lines intersect at a point, infinite solutions if they are the same line, and no solution if they are parallel.
Applications
Systems of equations solve mixture problems, speed and time problems, costs and revenues in business, electrical circuits (Kirchhoff's laws), and force equilibrium in mechanics.