 48.1: The average of two numbers is 10. What is the sum of the twonumbers?
 48.2: What is the cost of 10.0 gallons of gasoline priced at $2.279 per g...
 48.3: The movie started at 11:45 a.m. and ended at 1:20 p.m. The movie wa...
 48.4: Three of the numbers shown below are equal. Which numberis not equa...
 48.5: Arrange these numbers in order from least to greatest: 1.02, 0.102,...
 48.6: 0.1 + 0.2 + 0.3 + 0.4
 48.7: (8)(0.125)
 48.8: Juan was hiking to a waterfall 3 miles away. After hiking2.1 miles,...
 48.9: Estimate the sum of 4967, 8142, and 6890.
 48.10: 8 0.144
 48.11: 6 0.9
 48.12: 4 0.9
 48.13: What is the cost of 100 pens priced at 39 each?
 48.14: Write (5 10) (6 110) (4 1100) in standard form.
 48.15: What is the least common multiple of 6 and 8?
 48.16: Find each unknown number:
 48.17: 12 m 523
 48.18: n 234 514
 48.19: x + 3.21 = 4
 48.20: What fraction is 23 of 34?
 48.21: Sam carried the football three times during the game. He had gains ...
 48.22: If a spool for thread is 2 cm in diameter, then one wind ofthread i...
 48.23: If a rectangle is 12 inches long and 8 inches wide, what is the rat...
 48.24: The perimeter of this square is 4 feet. What isits perimeter in inc...
 48.25: The area of this square is one square foot.What is its area in squa...
 48.26: If d = rt, and if r = 60 and t = 4,what does d equal?
 48.27: Seventyfive percent of the 32 chairs in the room were occupied. Wr...
 48.28: Rename 13 and 14 as fractions with denominators of 12. Then add the...
 48.29: Multiply as shown. Then simplify the answer.3.50.7 1010
 48.30: There were 156 316 pies on the shelf. How can the server take 156 3...
Solutions for Chapter 48: Subtracting Mixed Numbers with Regrouping, Part 1
Full solutions for Saxon Math, Course 1  1st Edition
ISBN: 9781591417835
Solutions for Chapter 48: Subtracting Mixed Numbers with Regrouping, Part 1
Get Full SolutionsThis textbook survival guide was created for the textbook: Saxon Math, Course 1, edition: 1. Since 30 problems in chapter 48: Subtracting Mixed Numbers with Regrouping, Part 1 have been answered, more than 37886 students have viewed full stepbystep solutions from this chapter. Saxon Math, Course 1 was written by and is associated to the ISBN: 9781591417835. Chapter 48: Subtracting Mixed Numbers with Regrouping, Part 1 includes 30 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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).

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.

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.

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

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

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 ().

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.