Solving Systems by Substitution: Detailed Step-by-Step Guide
The substitution method is one of the primary algebraic techniques for solving systems of simultaneous linear equations. It works by converting a two-variable system into a single one-variable equation.
The 4-Step Substitution Protocol
- 1Isolate a variable: Choose one equation and isolate either variable with coefficient $\pm 1$ if possible:
y = \frac{c_1 - a_1 x}{b_1}
- 1Substitute: Replace that variable in the other equation with the newly formed algebraic expression.
- 2Solve for the single variable: Simplify the equation to solve for $x$.
- 3Back-substitute: Insert the found value of $x$ back into the isolated equation from Step 1 to determine $y$.
System Classification Matrix
| Determinant | Equations Relationship | Geometric Picture | Number of Solutions |
|---|---|---|---|
| $D \ne 0$ | Consistent & Independent | Lines intersect at one point | Unique $(x, y)$ |
| $D = 0, D_x \ne 0$ | Inconsistent | Parallel lines | No solution |
| $D = 0, D_x = 0$ | Consistent & Dependent | Coincident (same line) | Infinitely many |