Systems of Equations and Inequalities

At the start of the Second World War, British military and intelligence officers recognized that defeating Nazi Germany would require the Allies to know what the enemy was planning — a task complicated by the Enigma machine, whose rotors and plugboard let German encoders add complex encryption to everyday military communications. Poland made the first significant advances by recruiting math students from Poznań University as codebreakers, and the Allied effort that followed turned on mathematics. In this chapter, we investigate the kind of mathematics that underlies work with many interacting unknowns at once: systems of equations and inequalities, and the matrix methods that solve them systematically.

Sections

• Systems of Linear Equations: Two Variables — solve systems by graphing, substitution, and the addition method, identify inconsistent and dependent systems, and analyze cost, revenue, and break-even applications.
• Systems of Linear Equations: Three Variables — solve 3×3 systems by elimination and back-substitution, classify inconsistent and dependent systems, and model applications with three unknowns.
• Systems of Nonlinear Equations and Inequalities: Two Variables — solve systems pairing lines, parabolas, circles, and ellipses by substitution and elimination, and graph nonlinear inequalities and their systems’ feasible regions.
• Partial Fractions — decompose rational expressions over nonrepeated and repeated linear factors and over nonrepeated and repeated irreducible quadratic factors.
• Matrices and Matrix Operations — find matrix dimensions, add, subtract, and scalar-multiply matrices, and compute matrix products, including applied cost tables.
• Solving Systems with Gaussian Elimination — write augmented matrices, perform row operations, reduce to row-echelon form, and solve 2×2 and 3×3 systems, including finance applications.
• Solving Systems with Inverses — verify and find multiplicative inverses of 2×2 and 3×3 matrices by formula and by augmenting with the identity, and solve systems as matrix equations.
• Solving Systems with Cramer’s Rule — evaluate 2×2 and 3×3 determinants, solve systems with Cramer’s Rule, recognize inconsistent and dependent cases, and use determinant properties.

The sections above are adapted from Precalculus 2e, Chapter 9 by Jay Abramson, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. The chapter opener’s Enigma-machine photograph is omitted; its historical framing is condensed above.