One Solution, No Solution, and Infinitely Many Solutions

Objective: Recognize a nonzero remaining coefficient.

Prerequisite skills: expanding expressions and collecting like terms. Simplifying ax + b = cx + d gives (a − c)x = d − b. If a − c is nonzero, exactly one real x works. The variable does not disappear. Common error: assuming any equation with x on both sides has no solution. Restrictions and interpretation: Classify after simplification; never divide by a coefficient that becomes zero.

Objective: Distinguish no solutions from infinitely many.

If all variable terms cancel, inspect the constants. A false statement such as 0 = 5 is a contradiction and has no solutions. A true statement such as 0 = 0 is an identity and holds for every value allowed by the original equation. Common error: treating cancellation of x as proof of infinitely many solutions. Restrictions and interpretation: Classify after simplification; never divide by a coefficient that becomes zero.

Objective: Choose a constant that makes an equation an identity.

An equation is true for all real values precisely when its variable coefficients and constant terms match after expansion. Matching only the variable terms can leave a contradiction. Common error: matching coefficients while ignoring constants. Restrictions and interpretation: Classify after simplification; never divide by a coefficient that becomes zero. 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.