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Unit A5: Systems and Simultaneous Constraints

Extend linear reasoning to multiple equations, variables, and feasible regions. Which values satisfy several requirements simultaneously?

Act V - Satisfying several conditions at once

The 8-lesson path

Intersections represent simultaneous truth. Graphing estimates solutions, substitution replaces equals by equals, elimination combines equations to remove a variable, and applications reveal when two variables are the natural model.

Governing question

Which values satisfy several requirements simultaneously?

Extend linear reasoning to multiple equations, variables, and feasible regions.

Mastery design

Know when to continue

Mixed systems require method selection and interpretation of the ordered-pair solution.

Unit outcomes
  1. Interpret a system solution as satisfying every constraint.
  2. Solve systems graphically, by substitution, and by elimination.
  3. Classify one, zero, or infinitely many solutions.
  4. Model totals, mixtures, rates, and break-even situations.
  5. Graph systems of inequalities and interpret feasible regions.
Core sequence

Lessons

Move in order when this material is new; use prerequisites and repair links when entering mid-unit.

  1. 01Systems as simultaneous conditionsUnderstand a solution as an ordered pair satisfying every equation or constraint.
  2. 02Graphical solutionsRead intersections and estimate exact or approximate system solutions.
  3. 03SubstitutionReplace one variable expression with an equal expression to reduce a system.
  4. 04EliminationCombine scaled equations to remove a variable while preserving the common solution.
  5. 05Dependent and inconsistent systemsConnect algebraic outcomes 0=00=0 and 0=c0=c to coincident and parallel lines.
  6. 06Applications of systemsModel problems whose structure genuinely requires two unknown quantities.
  7. 07Systems of inequalitiesFind regions satisfying multiple linear bounds and distinguish included boundaries.
  8. 08Systems strategy studioChoose graphing, substitution, or elimination based on structure and purpose.

Practice around the path

Review, mixed practice, mastery, and investigation

Every route contains a fixed set of concrete questions, with detailed solutions available after an attempt.

ReviewCumulative reviewRevisit this unit with prior-unit retrieval.PracticeMixed practiceClassify the structure before choosing a method.MasteryMastery checkUse the supplied grading and cumulative-share contract.InvestigationPerformance taskDesign a production or allocation plan subject to budget, time, and minimum-output constraints.Response guideMastery response guideAttempt-gated rubric support; no invented canonical answers.

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