1.7 Modeling with Equations
Focus on translate and solve and distance model.
Before you solve
Ideas in this lesson
Review the ideas below. Select one to see worked steps and five practice questions.
01Expressing quantities with a variableWorked steps · five questions02Money, costs and averagesWorked steps · five questions03Interest and investmentsWorked steps · five questions04Geometry and dimensionsWorked steps · five questions05Mixture and concentrationWorked steps · five questions06Work problemsWorked steps · five questions07Distance, rate and timeWorked steps · five questions08Number, consecutive-integer and age problemsWorked steps · five questions09Solid geometry and balanceWorked steps · five questions10Linked geometry, paths and model communicationWorked steps · five questions
Lesson brief and rules
1.7 Modeling with Equations
Define a variable, translate the situation into an equation, and check the unit.
Worked example
01 / A theater sells tickets. Adult tickets cost dollars and child tickets cost dollars. If is the number of adult tickets, which expression models total revenue and what constraint is required?
02 / Define as the adult-ticket count, so the child-ticket count is .
03 / Adult revenue is ; child revenue is .
04 / Add the two revenues, because revenue is the conserved total.
05 / Both counts are nonnegative integers, so must be an integer between and the total.
Worked example 2 · Expressing quantities with a variable
A cyclist travels at km/h, then returns over the same route at a speed km/h faster. Which expression is the total travel time, with a valid domain?
expresses each leg’ time.
The outward speed is , and the return speed is .
Add the separate leg times because the trips occur consecutively.
The outward speed must be positive; that also makes the faster return speed positive.
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.