Substitution and Elimination

Both methods do the same thing: get rid of one variable so a single equation remains.

Substitution

If one equation is already solved for a variable, put that expression into the other.

Elimination

Add or subtract the equations so one variable cancels. Multiply first if needed.

Which to use

Substitution suits an equation already solved for a variable; elimination suits matching coefficients.

Two algebraic routes

Substitution replaces one variable with an expression in the other. Elimination adds or subtracts the equations so one variable cancels. Both reduce a two-variable system to a single equation in one variable.

Which to choose

Use substitution when a variable is already isolated or has coefficient 1. Use elimination when the equations are in standard form with matching or easily matched coefficients. Both always work; one is usually much cleaner.

The three outcomes

One solution when the lines cross, none when they are parallel, infinitely many when they coincide. Algebraically, the last two show up as a false or true numerical statement after both variables cancel.

Check in both original equations

A solution must satisfy both. Checking only one proves nothing, since every point on one line satisfies that equation. This is the most frequently skipped and most frequently needed check in the chapter.