Chapter 10: Systems of Equations and Inequalities
Systems of equations require multiple equations satisfied simultaneously. We solve them by substitution, elimination, and matrix methods. This chapter covers linear systems, matrices, determinants, inverse matrices, nonlinear systems, and linear programming ā essential tools for algebra, calculus, economics, and data science.
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Sections
10.1Systems of Linear Equations in Two Variables
Two linear equations in two unknowns describe two lines. Their solution is the intersection point. Three possibilities: one solution (intersecting), no solution (parallel), infinitely many solutions (same line). Methods include graphing, substitution, and elimination.
10.2Systems of Linear Equations in Several Variables
Three equations in three unknowns are solved by Gaussian elimination ā reducing an augmented matrix to row echelon form and back-substituting. This is the foundation of matrix methods used in data science and engineering.
10.3Matrices and Operations on Matrices
A matrix is a rectangular array of numbers. Matrix operations include addition, scalar multiplication, and matrix multiplication. Matrix multiplication is NOT commutative ā order matters. Determinants measure a matrix's 'scaling factor' and determine whether an inverse exists.
10.4Nonlinear Systems and Inequalities
Nonlinear systems (circles, parabolas, and lines) can have 0, 1, or 2 solutions. Systems of inequalities define feasible regions in the plane. Linear programming finds the optimal value of a linear objective function over such a region ā the maximum or minimum always occurs at a corner point.
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