The Substitution Method
The substitution method is a technique for solving systems of linear equations. It involves solving one equation for one variable, then substituting that expression into the other equation to find the remaining variable.
Steps of the Substitution Method
Step 1: Isolate
Choose one equation and solve for one variable (x or y) in terms of the other.
ax + by = c => x = (c - by)/a
Step 2: Substitute
Replace that variable in the other equation with the expression found in Step 1.
Plug expression into Eq. 2
Step 3: Solve & Back-substitute
Solve for the remaining variable, then substitute back to find the first variable.
Find y, then x = (c - by)/a
System Types
- Consistent Independent: Exactly one solution (lines intersect at one point).
- Consistent Dependent: Infinitely many solutions (same line).
- Inconsistent: No solution (parallel lines).
When to Use Substitution
The substitution method is especially convenient when one of the coefficients is 1 or -1, making it easy to isolate a variable. For larger systems or more complex coefficients, elimination or matrix methods may be more efficient.