 42.1: Write 12 and 23 as fractions with denominators of 6. Thenadd the re...
 42.2: According to some estimates, our own galaxy, the Milky Way, contain...
 42.3: The rectangular school yard is 120 yards long and 40 yardswide. How...
 42.4: What number is 40% of 30?
 42.5: In problems 5 and 6, multiply 12 by a fraction equal to 1 to comple...
 42.6: In problems 5 and 6, multiply 12 by a fraction equal to 1 to comple...
 42.7: 4.32 0.6 281
 42.8: 6.3 0.54
 42.9: (0.15)2
 42.10: What is the reciprocal of 67?
 42.11: Which digit in 12,345 has the same place value as the 6 in 67.89?
 42.12: What is the least common multiple of 3, 4, and 6?
 42.13: 535 445
 42.14: 236 423
 42.15: 83 12
 42.16: 65 3
 42.17: 1 14
 42.18: 1010 55
 42.19: Form three different fractions that are equal to 13 . (Hint: Multip...
 42.20: The prime numbers that multiply to form 35 are 5 and 7.Which prime ...
 42.21: In three games Almas scores were 12,143; 9870; and 14,261.Describe ...
 42.22: Estimate the quotient of 817641 . Describe how you performed theest...
 42.23: How many eggs are in 23 of a dozen? Draw a diagram toillustrate the...
 42.24: Write 34 with a denominator of 8. Subtract the renamed fractionfrom...
 42.25: What is the perimeter of this rectangle?
 42.26: What is the area of this rectangle?
 42.27: The regular price r minus the discount d equals the saleprice sr d ...
 42.28: Below we show the same division problem written three different way...
 42.29: What time is 212 hours after 11:45 a.m.?
 42.30: a. How many 56 s are in 1?b. Use the answer to part a to find the n...
Solutions for Chapter 42: Renaming Fractions by Multiplying by 1
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Solutions for Chapter 42: Renaming Fractions by Multiplying by 1
Get Full SolutionsChapter 42: Renaming Fractions by Multiplying by 1 includes 30 full stepbystep solutions. This textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. Since 30 problems in chapter 42: Renaming Fractions by Multiplying by 1 have been answered, more than 35180 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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 .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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).

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.

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.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.