 11.1.1: Solve the equation: 3x + 4 = 8  x. (pp. A43A44)
 11.1.2: (a) Graph the line: 3x + 4y = 12. (b) What is the slop
 11.1.3: If a system of equations has no solution, it is said to be
 11.1.4: If a system of equations has one solution, the system is and the eq...
 11.1.5: If the solution to a system of two linear equations containing two ...
 11.1.6: If the lines that make up a system of two linear equations are coin...
 11.1.7: In 716, verify that the values of the variables listed are solution...
 11.1.8: In 716, verify that the values of the variables listed are solution...
 11.1.9: In 716, verify that the values of the variables listed are solution...
 11.1.10: In 716, verify that the values of the variables listed are solution...
 11.1.11: In 716, verify that the values of the variables listed are solution...
 11.1.12: In 716, verify that the values of the variables listed are solution...
 11.1.13: In 716, verify that the values of the variables listed are solution...
 11.1.14: In 716, verify that the values of the variables listed are solution...
 11.1.15: In 716, verify that the values of the variables listed are solution...
 11.1.16: In 716, verify that the values of the variables listed are solution...
 11.1.17: In 1754, solve each system of equations. If the system has no solut...
 11.1.18: In 1754, solve each system of equations. If the system has no solut...
 11.1.19: In 1754, solve each system of equations. If the system has no solut...
 11.1.20: In 1754, solve each system of equations. If the system has no solut...
 11.1.21: In 1754, solve each system of equations. If the system has no solut...
 11.1.22: In 1754, solve each system of equations. If the system has no solut...
 11.1.23: In 1754, solve each system of equations. If the system has no solut...
 11.1.24: In 1754, solve each system of equations. If the system has no solut...
 11.1.25: In 1754, solve each system of equations. If the system has no solut...
 11.1.26: In 1754, solve each system of equations. If the system has no solut...
 11.1.27: In 1754, solve each system of equations. If the system has no solut...
 11.1.28: In 1754, solve each system of equations. If the system has no solut...
 11.1.29: In 1754, solve each system of equations. If the system has no solut...
 11.1.30: In 1754, solve each system of equations. If the system has no solut...
 11.1.31: In 1754, solve each system of equations. If the system has no solut...
 11.1.32: In 1754, solve each system of equations. If the system has no solut...
 11.1.33: In 1754, solve each system of equations. If the system has no solut...
 11.1.34: In 1754, solve each system of equations. If the system has no solut...
 11.1.35: In 1754, solve each system of equations. If the system has no solut...
 11.1.36: In 1754, solve each system of equations. If the system has no solut...
 11.1.37: In 1754, solve each system of equations. If the system has no solut...
 11.1.38: In 1754, solve each system of equations. If the system has no solut...
 11.1.39: In 1754, solve each system of equations. If the system has no solut...
 11.1.40: In 1754, solve each system of equations. If the system has no solut...
 11.1.41: In 1754, solve each system of equations. If the system has no solut...
 11.1.42: In 1754, solve each system of equations. If the system has no solut...
 11.1.43: In 1754, solve each system of equations. If the system has no solut...
 11.1.44: In 1754, solve each system of equations. If the system has no solut...
 11.1.45: In 1754, solve each system of equations. If the system has no solut...
 11.1.46: In 1754, solve each system of equations. If the system has no solut...
 11.1.47: In 1754, solve each system of equations. If the system has no solut...
 11.1.48: In 1754, solve each system of equations. If the system has no solut...
 11.1.49: In 1754, solve each system of equations. If the system has no solut...
 11.1.50: In 1754, solve each system of equations. If the system has no solut...
 11.1.51: In 1754, solve each system of equations. If the system has no solut...
 11.1.52: In 1754, solve each system of equations. If the system has no solut...
 11.1.53: In 1754, solve each system of equations. If the system has no solut...
 11.1.54: In 1754, solve each system of equations. If the system has no solut...
 11.1.55: The perimeter of a rectangular floor is 90 feet. Find the dimension...
 11.1.56: The length of fence required to enclose a rectangular field is 3000...
 11.1.57: In 2005 there was a total of 55 commercial and noncommercial orbita...
 11.1.58: A movie theater charges $9.00 for adults and $7.00 for senior citiz...
 11.1.59: A store sells cashews for $5.00 per pound and peanuts for $1.50 per...
 11.1.60: A recently retired couple needs $12,000 per year to supplement thei...
 11.1.61: With a tail wind, a small Piper aircraft can fly 600 miles in 3 hou...
 11.1.62: The average airspeed of a singleengine aircraft is 150 miles per ho...
 11.1.63: A restaurant manager wants to purchase 200 sets of dishes. One desi...
 11.1.64: One group of people purchased 10 hot dogs and 5 soft drinks at a co...
 11.1.65: he grocery store we use does not mark prices on its goods. My wife ...
 11.1.66: Pamela requires 3 hours to swim 15 miles downstream on the Illinois...
 11.1.67: A doctors prescription calls for a daily intake containing 40 milli...
 11.1.68: A doctors prescription calls for the creation of pills that contain...
 11.1.69: Find real numbers a, b, and c so that the graph of the function y =...
 11.1.70: Find real numbers a, b, and c so that the graph of the function y =...
 11.1.71: In economics, the IS curve is a linear equation that represents all...
 11.1.72: In economics, the IS curve is a linear equation that represents all...
 11.1.73: An application of Kirchhoffs Rules to the circuit shown below resul...
 11.1.74: An application of Kirchhoffs Rules to the circuit shown results in ...
 11.1.75: A Broadway theater has 500 seats, divided into orchestra, main, and...
 11.1.76: A movie theater charges $8.00 for adults, $4.50 for children, and $...
 11.1.77: A dietitian wishes a patient to have a meal that has 66 grams (g) o...
 11.1.78: Kelly has $20,000 to invest. As her financial planner, you recommen...
 11.1.79: One group of customers bought 8 deluxe hamburgers, 6 orders of larg...
 11.1.80: Use the information given in 79. Suppose that a third group purchas...
 11.1.81: Use the information given in 79. Suppose that a third group purchas...
 11.1.82: Make up a system of three linear equations containing three variabl...
 11.1.83: Write a brief paragraph outlining your strategy for solving a syste...
 11.1.84: Do you prefer the method of substitution or the method of eliminati...
Solutions for Chapter 11.1: Systems of Linear Equations: Substitution and Elimination
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 11.1: Systems of Linear Equations: Substitution and Elimination
Get Full SolutionsChapter 11.1: Systems of Linear Equations: Substitution and Elimination includes 84 full stepbystep solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Since 84 problems in chapter 11.1: Systems of Linear Equations: Substitution and Elimination have been answered, more than 56208 students have viewed full stepbystep solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.