 41.1: Find the rate of change represented in each table or graph.
 41.2: Find the rate of change represented in each table or graph.
 41.3: SPORTS For Exercises 35, use the graph at the right.
 41.4: SPORTS For Exercises 35, use the graph at the right.
 41.5: SPORTS For Exercises 35, use the graph at the right.
 41.6: Find the slope of the line that passes through each pair of points.
 41.7: Find the slope of the line that passes through each pair of points.
 41.8: Find the slope of the line that passes through each pair of points.
 41.9: Find the slope of the line that passes through each pair of points.
 41.10: Find the slope of the line that passes through each pair of points.
 41.11: Find the slope of the line that passes through each pair of points.
 41.12: Find the value of r so the line that passes through each pair of po...
 41.13: Find the value of r so the line that passes through each pair of po...
 41.14: Find the rate of change represented in each table or graph.
 41.15: Find the rate of change represented in each table or graph.
 41.16: Find the rate of change represented in each table or graph.
 41.17: Find the rate of change represented in each table or graph.
 41.18: SPORTS What was the annual rate of change Women Competing in Triath...
 41.19: CELL PHONES In 2000, 5% of 13 to 17yearolds had cell phones. By ...
 41.20: Find the slope of the line that passes through each pair of points.
 41.21: Find the slope of the line that passes through each pair of points.
 41.22: Find the slope of the line that passes through each pair of points.
 41.23: Find the slope of the line that passes through each pair of points.
 41.24: Find the slope of the line that passes through each pair of points.
 41.25: Find the slope of the line that passes through each pair of points.
 41.26: Find the slope of the line that passes through each pair of points.
 41.27: Find the slope of the line that passes through each pair of points.
 41.28: Find the slope of the line that passes through each pair of points.
 41.29: Find the slope of the line that passes through each pair of points.
 41.30: Find the slope of the line that passes through each pair of points.
 41.31: Find the slope of the line that passes through each pair of points.
 41.32: Find the value of r so the line that passes through each pair of po...
 41.33: Find the value of r so the line that passes through each pair of po...
 41.34: Find the value of r so the line that passes through each pair of po...
 41.35: Find the value of r so the line that passes through each pair of po...
 41.36: Find the slope of the line that passes through each pair of points.
 41.37: Find the slope of the line that passes through each pair of points.
 41.38: Find the slope of the line that passes through each pair of points.
 41.39: DRIVING When driving up a certain hill, you rise 15 feet for every ...
 41.40: CONSTRUCTION Use a ruler to estimate the slope of each object.
 41.41: CONSTRUCTION Use a ruler to estimate the slope of each object.
 41.42: CONSTRUCTION Use a ruler to estimate the slope of each object.
 41.43: CONSTRUCTION Use a ruler to estimate the slope of each object.
 41.44: Find the value of r so the line that passes through each pair of po...
 41.45: Find the value of r so the line that passes through each pair of po...
 41.46: Find the value of r so the line that passes through each pair of po...
 41.47: Find the value of r so the line that passes through each pair of po...
 41.48: ANALYZE TABLES For Exercises 4850, use the table that shows Karens ...
 41.49: ANALYZE TABLES For Exercises 4850, use the table that shows Karens ...
 41.50: ANALYZE TABLES For Exercises 4850, use the table that shows Karens ...
 41.51: ANALYZE GRAPHS For Exercises 5153, 1*LVV i >`iq x n nx x Li { x {...
 41.52: ANALYZE GRAPHS For Exercises 5153, 1*LVV i >`iq x n nx x Li { x {...
 41.53: ANALYZE GRAPHS For Exercises 5153, 1*LVV i >`iq x n nx x Li { x {...
 41.54: GROWTH RATE For Exercises 5456, use the following information. Afte...
 41.55: GROWTH RATE For Exercises 5456, use the following information. Afte...
 41.56: GROWTH RATE For Exercises 5456, use the following information. Afte...
 41.57: CONSTRUCTION The slope of a stairway tread (ideal riser (ideal 7 in...
 41.58: RESEARCH Use the Internet or another reference to find the populati...
 41.59: CHALLENGE Develop a strategy for determining whether the slope of t...
 41.60: OPEN ENDED Integrate what you know Time (wk) Height of Plant (in.) ...
 41.61: CHALLENGE Determine whether Q(2, 3), R(1, 1), and S(4, 2) lie o...
 41.62: FIND THE ERROR Carlos and Allison are finding the slope of the line...
 41.63: Writing in Math Discuss how to find the slope of a roof and compare...
 41.64: A music store has x CDs in stock. If 350 are sold and 3y are added ...
 41.65: REVIEW A recipe for fruit punch calls for 2 ounces of orange juice ...
 41.66: Write an equation in function notation for each relation.
 41.67: Write an equation in function notation for each relation.
 41.68: Find the next three terms of each arithmetic sequence.
 41.69: Find the next three terms of each arithmetic sequence.
 41.70: Find the next three terms of each arithmetic sequence.
 41.71: Find the next three terms of each arithmetic sequence.
 41.72: FOOD Garrett is making _1 3 pound hamburgers. One pound of hamburg...
 41.73: PREREQUISITE SKILL Find each quotient.
 41.74: PREREQUISITE SKILL Find each quotient.
 41.75: PREREQUISITE SKILL Find each quotient.
 41.76: PREREQUISITE SKILL Find each quotient.
Solutions for Chapter 41: Steepness of a Line
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 41: Steepness of a Line
Get Full SolutionsAlgebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Chapter 41: Steepness of a Line includes 76 full stepbystep solutions. Since 76 problems in chapter 41: Steepness of a Line have been answered, more than 33957 students have viewed full stepbystep solutions from this chapter.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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