Substitution Method
AlgebraThe 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 taking one equation, solving it for one variable, and then "substituting" (plugging) that expression into the other equation. This gives you one equation with one variable.
Example
$y = 2x$ and $x + y = 9$. Substitute $2x$ for $y$ in the second: $x + 2x = 9$, so $3x = 9$, $x = 3$. Then $y = 2(3) = 6$. Solution: $(3, 6)$.
Key Insight
Substitution is like replacement: once you know what one variable equals, you replace it everywhere with that expression to reduce the problem to one equation.
Definition
In substitution, isolate one variable in one equation (choose the variable with coefficient $1$ for ease), then substitute the resulting expression into the second equation. Solve the single-variable equation, then back-substitute to find the other variable. Works well when one equation is already solved for a variable.
Example
$3x - y = 4$ and $2x + 3y = 19$. From equation 1: $y = 3x - 4$. Substitute into equation 2: $2x + 3(3x - 4) = 19$, so $2x + 9x - 12 = 19$, $11x = 31$, $x = 31/11$. Then $y = 3(31/11) - 4 = 93/11 - 44/11 = 49/11$.
Key Insight
Substitution is particularly powerful when one equation has a coefficient of $1$, making it easy to isolate a variable without fractions. In nonlinear systems, substitution is often the only practical algebraic method.
Definition
Substitution is a special case of variable elimination in which one equation is solved explicitly for a variable to produce a substitution mapping. For linear systems, substitution and elimination are equivalent in computational complexity. For nonlinear systems such as polynomial systems, substitution enables reduction to a single-variable polynomial, whose roots can then be found by resultants or Grobner bases.
Example
In the nonlinear system $x^2 + y = 5$ and $y = x + 1$, substitute to get $x^2 + x + 1 = 5$, so $x^2 + x - 4 = 0$. Solutions: $x = (-1 \pm \sqrt{17})/2$.
Key Insight
Substitution underpins the method of Lagrange multipliers in constrained optimization: solve the constraint for one variable and substitute into the objective function to reduce dimension.