Solving Systems by Substitution

Objective: Substitute an isolated variable into another equation.

Prerequisite skills: solving linear equations and substituting expressions. If an equation already gives y in terms of x, replace y with that entire expression in the other equation. Solve for x, then use either original equation to find y. Common error: substituting just the coefficient instead of the complete expression. Restrictions and interpretation: A solution is an ordered pair satisfying both original equations.

Objective: Rewrite one equation to make substitution efficient.

You may solve either equation for either variable. Prefer a coefficient of 1 or −1 to avoid introducing unnecessary fractions. Parenthesize the expression before multiplying it. Common error: forgetting to distribute a coefficient across the replacement expression. Restrictions and interpretation: A solution is an ordered pair satisfying both original equations.

Objective: Find a combination of the solution coordinates.

Some questions ask for x + y or another expression rather than one coordinate. Solve the system or exploit a direct relationship, then evaluate that exact target. Checking only one equation is insufficient. Common error: reporting x when the target combines x and y. Restrictions and interpretation: A solution is an ordered pair satisfying both original equations. 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.