substitution method

Suppose one equation tells you, in effect, that y is the same thing as 2x + 1. Then wherever another equation mentions y, you can swap in 2x + 1 in its place — you have substituted a known expression for the variable. The substitution method turns a two-variable problem into a one-variable problem you already know how to solve.

The recipe: solve one of the equations for one variable in terms of the other, then substitute that expression into the remaining equation. That second equation now contains only a single variable, so you solve it normally. Finally, plug the value you found back into your expression to recover the other variable.

Substitution shines when one variable already stands alone or is easy to isolate — for instance when an equation reads y = something. When every variable comes wrapped in awkward coefficients, the elimination method is often tidier. Either way, always verify your pair in both original equations, because an arithmetic slip during substitution is easy to make and easy to catch.