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BetterGrades Precalculus · Unit 8 · Unit map

Systems, Matrices, and Multivariable Models

A system solution satisfies every equation simultaneously and appears graphically as a common intersection.

Precalculus · Unit 8

The lesson path

A system solution satisfies every equation simultaneously and appears graphically as a common intersection.

Core sequence

Systems, Matrices, and Multivariable Models

Move in order when the material is new, or enter at the exact skill you need.

  1. 01Systems as intersections and modelsInterpret a system solution as an input-output point satisfying every equation and solve two-variable systems graphically and algebraically.
  2. 02Nonlinear systemsSolve systems involving lines, quadratics, circles, rational, exponential, or other nonlinear functions and interpret all real intersections.
  3. 03Systems of inequalities and feasible regionsGraph simultaneous inequalities, identify feasible regions, and interpret boundaries and vertices.
  4. 04Systems in three variablesSolve and classify three-variable linear systems and interpret their solutions as intersections of planes.
  5. 05Matrix notation and operationsRepresent rectangular arrays with matrices and perform dimension-compatible addition, scalar multiplication, and matrix multiplication.
  6. 06Augmented matrices and row operationsTranslate linear systems into augmented matrices and justify elementary row operations as equivalent-system transformations.
  7. 07Gaussian elimination and echelon formUse forward elimination to reach row-echelon form and solve by back-substitution.
  8. 08Gauss-Jordan elimination and reduced row-echelon formReduce matrices to RREF and interpret pivots, free variables, rank, and consistency directly.
  9. 09Determinants, singularity, and inverse matricesCompute and interpret determinants, identify singular matrices, and solve suitable systems with inverse matrices.
  10. 10Linear transformations and multivariable modelingInterpret matrices as transformations of vectors and construct a complete multivariable model using equations, inequalities, and matrices.
Source record

References used for this unit

  • Yoshiwara, Modeling, Functions, and Graphs
  • Utah College Algebra
  • Stitz and Zeager, Precalculus

The learner copy is original BetterGrades material informed by these rights-separated references; no long source passage is reproduced.