Linear Equation Word Problems

Objective: Model a fixed starting amount and repeated equal additions.

Prerequisite skills: translating expressions and solving linear equations. A quantity with a constant change can be represented as start + rate × number of intervals. Solve for the unknown count by removing the starting amount and dividing by the per-interval change. Common error: dividing the final total without first removing the initial amount. Restrictions and interpretation: Define the unknown and check nonnegative or integer constraints when the context requires them.

Objective: Set equal two expressions that describe the same total.

To locate a break-even quantity, represent each plan with the same variable and equate the costs. A negative algebraic answer can mean there is no feasible break-even point within the context. Common error: comparing rates without including the fixed charges. Restrictions and interpretation: Define the unknown and check nonnegative or integer constraints when the context requires them.

Objective: Express related unknown quantities with one variable.

When one quantity differs from another by a known amount, write it in terms of one unknown. Substitute into the total constraint and solve. The final question may ask for the other quantity. Common error: solving for the smaller quantity when the question asks for the larger. Restrictions and interpretation: Define the unknown and check nonnegative or integer constraints when the context requires them. 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.