Substitution Method
The substitution method solves a system of equations by solving one equation for a variable and substituting that expression into the other equation.
Definition
The substitution method solves a system by isolating one variable in one equation (ideally one with coefficient $1$, to avoid fractions), then substituting that expression into the other equation to reduce the problem to a single equation in one variable, which is then solved and used to back-substitute for the other variable. It is especially useful when one equation is already solved for a variable, and in nonlinear systems it is often the only practical algebraic method: substitution is a special case of variable elimination that reduces a system to a single-variable polynomial, whose roots can then be found by resultants or Grobner bases.
Example
For $y = 2x$ and $x + y = 9$, substituting gives $x + 2x = 9$, so $x = 3$ and $y = 6$. For $3x - y = 4$ and $2x + 3y = 19$, solve the first for $y = 3x - 4$, substitute into the second to get $11x = 31$, so $x = 31/11$ and $y = 49/11$. In the nonlinear system $x^2 + y = 5$ and $y = x + 1$, substituting gives $x^2 + x - 4 = 0$, with solutions $x = (-1 \pm \sqrt{17})/2$.
Key Insight
Substitution is like replacement: once you know what one variable equals, you swap it in everywhere to shrink the problem to one equation. For linear systems, substitution and elimination are computationally equivalent, but substitution's real power shows in constrained optimization, where the method of Lagrange multipliers solves a constraint for one variable and substitutes it into the objective function to reduce the problem's dimension.