977 CHAPTER 9 Test Prep Chapter 9 Test Prep Key Terms 9.1 linear equation (firstdegree equation) in n unknowns system of equations solutions (of a system of equations) system of linear equations (linear system) consistent system independent equations inconsistent system dependent equations equivalent system ordered triple 9.2 matrix (matrices) element (of a matrix) augmented matrix dimension (of a matrix) 9.3 determinant minor cofactor expansion by a row or column Cramer’s rule 9.4 partial fraction decomposition partial fraction 9.5 nonlinear system 9.6 half-plane boundary linear inequality in two variables system of inequalities linear programming constraints objective function region of feasible solutions vertex (corner point) 9.7 square matrix of order n row matrix column matrix zero matrix additive inverse (negative) of a matrix scalar 9.8 identity matrix multiplicative inverse (of a matrix) New Symbols 1a, b, c2 ordered triple 3 A4 matrix A (graphing calculator symbolism) ∣ A∣ determinant of matrix A aij element in row i, column j of a matrix D, Dx, Dy determinants used in Cramer’s rule I2, I3 identity matrices A−1 multiplicative inverse of matrix A Quick Review Concepts Examples 9.1 Systems of Linear Equations Transformations of a Linear System 1. Interchange any two equations of the system. 2. Multiply or divide any equation of the system by a nonzero real number. 3. Replace any equation of the system by the sum of that equation and a multiple of another equation in the system. A system may be solved by the substitution method, the elimination method, or a combination of the two methods. Substitution Method Use one equation to find an expression for one variable in terms of the other, and then substitute this expression into the other equation of the system. Solve the system. 4 x - y = 7 (1) 3x + 2y = 30 (2) Solve for y in equation (1). y = 4x - 7 (3) Substitute 4x - 7 for y in equation (2), and solve for x. 3x + 214x - 72 = 30 (2) with y = 4x - 7 3x + 8x - 14 = 30 Distributive property 11x - 14 = 30 Combine like terms. 11x = 44 Add 14. x = 4 Divide by 11. Substitute 4 for x in the equation y = 4x - 7 to find that y = 9. The solution set is 514, 926.
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