Solving Systems by Elimination

Objective: Add equations whose variable coefficients are opposites.

Prerequisite skills: adding equations and multiplying by constants. Adding true equations gives a true equation. Opposite coefficients cancel a variable, leaving an equation in the other variable. Substitute back to recover the canceled variable. Common error: subtracting equations that should be added to cancel a term. Restrictions and interpretation: Scale every term of an equation before adding; retain signs when subtracting.

Objective: Multiply an equation to create canceling coefficients.

Choose multipliers that make one pair of variable coefficients equal in magnitude. Multiply every term, including the constant, then add or subtract consistently. Common error: multiplying the left side but leaving the right side unchanged. Restrictions and interpretation: Scale every term of an equation before adding; retain signs when subtracting.

Objective: Use equation sums to obtain a requested expression directly.

You do not always need both coordinates. If adding scaled equations produces the requested combination, solve for that combination directly. This is the same valid elimination principle with fewer steps. Common error: solving unnecessary coordinates or forgetting the common factor. Restrictions and interpretation: Scale every term of an equation before adding; retain signs when subtracting. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.