 13.1: Perform each indicated operation.1058  3110
 13.2: Perform each indicated operation.34,18
 13.3: Perform each indicated operation.5  142 + 122
 13.4: Perform each indicated operation.1322  14212425122  1223
 13.5: Perform each indicated operation.True or false?413  922  6 6
 13.6: Perform each indicated operation.Find the value of a xzwhen x = 2,...
 13.7: Perform each indicated operation.What property does 312 + x2 = 6 ...
 13.8: Perform each indicated operation.Simplify 4p  6 + 3p + 8 by combi...
 13.9: Solve.V = 13 pr2 h for h
 13.10: Solve.6  311 + x2 = 21x + 52  2
 13.11: Solve.1m  32 = 5  2m
 13.12: Solve.x  23 = 2x + 15
 13.13: Solve each inequality, and graph the solution set.2.5x 6 6.5
 13.14: Solve each inequality, and graph the solution set.41x + 32  5x 6 12
 13.15: Solve each inequality, and graph the solution set.23x  16 2
 13.16: Solve each problem.The gap in average annual earnings by level ofed...
 13.17: Solve each problem.Baby boomers are expected to inherit$10.4 trilli...
 13.18: Solve each problem.Use the answer from Exercise 17(b) to estimateth...
 13.19: Consider the linear equation Find the following.The x and yinterc...
 13.20: Consider the linear equation Find the following.The slope
 13.21: Consider the linear equation Find the following.The graph
 13.22: Are the lines with equations and parallel, perpendicular, or neither?
 13.23: Write an equation for each line. Give the final answer in slopeint...
 13.24: Write an equation for each line. Give the final answer in slopeint...
Solutions for Chapter 13: Linear Equations and Inequalities in Two Variables; Functions
Full solutions for Beginning Algebra  11th Edition
ISBN: 9780321673480
Solutions for Chapter 13: Linear Equations and Inequalities in Two Variables; Functions
Get Full SolutionsChapter 13: Linear Equations and Inequalities in Two Variables; Functions includes 24 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. Since 24 problems in chapter 13: Linear Equations and Inequalities in Two Variables; Functions have been answered, more than 37811 students have viewed full stepbystep solutions from this chapter. Beginning Algebra was written by and is associated to the ISBN: 9780321673480.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

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).

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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