 7.1.7.1.1: When two or more linear equations are considered simultaneously, th...
 7.1.7.1.2: The ordered pair or ordered pairs that satisfy all equations in a s...
 7.1.7.1.3: A system of equations that has no solution is called a(n) system of...
 7.1.7.1.4: A system of equations that has at least one solution is called a(n)...
 7.1.7.1.5: A system of equations that has an infinite number of solutions is c...
 7.1.7.1.6: If the graphs of the equations in a system of linear equations are ...
 7.1.7.1.7: If the graphs of the equations in a system of linear equations inte...
 7.1.7.1.8: If the graphs of the equations in a system of linear equations are ...
 7.1.7.1.9: A company makes a profit when its revenue is than its cost.
 7.1.7.1.10: A company has a loss when its revenue is than its cost.
 7.1.7.1.11: In Exercises 11 and 12, determine which ordered pairs are solutions...
 7.1.7.1.12: In Exercises 11 and 12, determine which ordered pairs are solutions...
 7.1.7.1.13: In Exercises 1316, solve the system of equations graphicallyx = 1 y...
 7.1.7.1.14: In Exercises 1316, solve the system of equations graphicallyx = 2 y...
 7.1.7.1.15: In Exercises 1316, solve the system of equations graphicallyx = 4 y...
 7.1.7.1.16: In Exercises 1316, solve the system of equations graphicallyx = 5 ...
 7.1.7.1.17: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.18: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.19: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.20: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.21: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.22: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.23: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.24: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.25: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.26: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.27: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.28: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.29: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.30: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.31: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.32: In Exercises 1732, solve the system of equations graphically. If th...
 7.1.7.1.33: a) If the two lines in a system of equations have different slopes,...
 7.1.7.1.34: Indicate whether the graph shown represents a consistent, inconsist...
 7.1.7.1.35: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.36: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.37: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.38: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.39: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.40: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.41: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.42: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.43: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.44: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.45: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.46: In Exercises 35 46, determine without graphing whether the system o...
 7.1.7.1.47: Two lines are perpendicular to each other when they meet at a right...
 7.1.7.1.48: Two lines are perpendicular to each other when they meet at a right...
 7.1.7.1.49: Two lines are perpendicular to each other when they meet at a right...
 7.1.7.1.50: Two lines are perpendicular to each other when they meet at a right...
 7.1.7.1.51: MODELINGTruck Rentals The cost of renting a mediumsized truck at U...
 7.1.7.1.52: MODELINGLandscaping Revisited In Example 5, assume that Toms Tree a...
 7.1.7.1.53: MODELINGSelling Backpacks Benjamins Backpacks can sell backpacks fo...
 7.1.7.1.54: MODELINGPurchasing Stocks When buying or selling stock for a custom...
 7.1.7.1.55: MODELINGManufacturing Bluray Disc Players A manufacturer sells Blu...
 7.1.7.1.56: Explain how you can determine whether a system of two linear equati...
 7.1.7.1.57: MODELINGJob Offers Hubert Hotchkiss had two job offers for sales po...
 7.1.7.1.58: MODELINGLongDistance Calling a) In September 2010, an AT&T One Rat...
 7.1.7.1.59: Points of Intersection a) If two lines have different slopes, what ...
 7.1.7.1.60: Connect all the following points using exactly four straightline se...
 7.1.7.1.61: The Rhind Papyrus The Rhind Papyrus indicates that the early Egypti...
Solutions for Chapter 7.1: Systems of Linear Equations and Inequalities
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 7.1: Systems of Linear Equations and Inequalities
Get Full SolutionsChapter 7.1: Systems of Linear Equations and Inequalities includes 61 full stepbystep solutions. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This expansive textbook survival guide covers the following chapters and their solutions. Since 61 problems in chapter 7.1: Systems of Linear Equations and Inequalities have been answered, more than 74066 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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!

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