 Chapter 3.1: Choose the term from the list above that best matches each phrase
 Chapter 3.2: Choose the term from the list above that best matches each phrase
 Chapter 3.3: Choose the term from the list above that best matches each phrase
 Chapter 3.4: Choose the term from the list above that best matches each phrase
 Chapter 3.5: Choose the term from the list above that best matches each phrase
 Chapter 3.6: Choose the term from the list above that best matches each phrase
 Chapter 3.7: Choose the term from the list above that best matches each phrase
 Chapter 3.8: Choose the term from the list above that best matches each phrase
 Chapter 3.9: Choose the term from the list above that best matches each phrase
 Chapter 3.10: Choose the term from the list above that best matches each phrase
 Chapter 3.11: Solve each system of linear equations by graphing.
 Chapter 3.12: Solve each system of linear equations by graphing.
 Chapter 3.13: Solve each system of linear equations by graphing.
 Chapter 3.14: Solve each system of linear equations by graphing.
 Chapter 3.15: Two plumbers offer competitive services. The first charges a $35 ho...
 Chapter 3.16: Solve each system of equations by using either substitution or elim...
 Chapter 3.17: Solve each system of equations by using either substitution or elim...
 Chapter 3.18: Solve each system of equations by using either substitution or elim...
 Chapter 3.19: Solve each system of equations by using either substitution or elim...
 Chapter 3.20: Solve each system of equations by using either substitution or elim...
 Chapter 3.21: Solve each system of equations by using either substitution or elim...
 Chapter 3.22: Colleen bought 15 used and lightly used Tshirts at a thrift store....
 Chapter 3.23: Solve each system of inequalities by graphing. Use a table to analy...
 Chapter 3.24: Solve each system of inequalities by graphing. Use a table to analy...
 Chapter 3.25: Solve each system of inequalities by graphing. Use a table to analy...
 Chapter 3.26: Solve each system of inequalities by graphing. Use a table to analy...
 Chapter 3.27: Tamara spends no more than 5 hours working at a local manufacturing...
 Chapter 3.28: Tamara spends no more than 5 hours working at a local manufacturing...
 Chapter 3.29: Solve each system of equations.
 Chapter 3.30: Solve each system of equations.
 Chapter 3.31: Solve each system of equations.
Solutions for Chapter Chapter 3: Solving Equations and Inequalities
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter Chapter 3: Solving Equations and Inequalities
Get Full SolutionsSince 31 problems in chapter Chapter 3: Solving Equations and Inequalities have been answered, more than 42679 students have viewed full stepbystep solutions from this chapter. Chapter Chapter 3: Solving Equations and Inequalities includes 31 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·