754 11 A Look Back In Appendix A and Chapters 1, 3, 4, 5, and 7 we solved various kinds of equations and inequalities involving a single variable. A Look Ahead In this chapter we take up the problem of solving equations and inequalities containing two or more variables. There are various ways to solve such problems. The method of substitution for solving equations in several variables dates back to ancient times. The method of elimination, although it had existed for centuries, was put into systematic order by Karl Friedrich Gauss (1777—1855) and by Camille Jordan (1838—1922). The theory of matrices was developed in 1857 by Arthur Cayley (1821—1895), although only later were matrices used as we use them in this chapter. Matrices are useful in almost all areas of mathematics. The method of determinants was invented by Takakazu Seki Kôwa (1642—1708) in 1683 in Japan and by Gottfried Wilhelm von Leibniz (1646—1716) in 1693 in Germany. Cramer’s Rule is named after Gabriel Cramer (1704—1752) of Switzerland, who popularized the use of determinants for solving linear systems. Section 11.5, on partial fraction decomposition , is an application of systems of equations and is used in integral calculus. Section 11.8 introduces linear programming , a modern application of linear inequalities. This topic is particularly useful for students interested in operations research. Economic Outcomes Annual Earnings of Young Adults Earnings increase with education: full-time workers with at least a bachelor’s degree have higher mean earnings than those with less education. Men and women who dropped out of high school earned 20% and 23% less, respectively, than high school graduates. The mean full-time earnings of young adults who had at least a bachelor’s degree declined in the 1970s relative to their counterparts who were high school graduates, before increasing between 1980 and 2019. Men with a bachelor’s degree or higher had earnings 19% higher than men high school graduates in 1980 and had earnings 92% higher in 2019. Among women, those with at least a bachelor’s degree had earnings 34% higher than women high school graduates in 1980, compared with earnings 83% higher in 2019. Source: U.S. Census Bureau —See Chapter Project— Outline 11. 1 Systems of Linear Equations: Substitution and Elimination 11. 2 Systems of Linear Equations: Matrices 11. 3 Systems of Linear Equations: Determinants 11. 4 Matrix Algebra 11. 5 Partial Fraction Decomposition 11. 6 Systems of Nonlinear Equations 11. 7 Systems of Inequalities 11. 8 Linear Programming Chapter Review Chapter Test Cumulative Review Chapter Project Systems of Equations and Inequalities Credit: Goodluz/Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=