 3.1: . The line graph shows the number, in millions,of real Christmas tr...
 3.2: Graph each line, using the given information or equation.2x + 3y = 12
 3.3: Graph each line, using the given information or equation.m = 1, 10,...
 3.4: Graph each line, using the given information or equation.y = 2x + 6
 3.5: Graph each line, using the given information or equation.m =  13, 42
 3.6: Graph each line, using the given information or equation.Undefined ...
 3.7: Graph each line, using the given information or equation.x  4y = 0
 3.8: Graph each line, using the given information or equation.y  4 = 9
 3.9: Graph each line, using the given information or equation.8x = 6y + 24
 3.10: Graph each line, using the given information or equation.m = 1, 10,...
 3.11: Graph each line, using the given information or equation.5x + 2y = 10
 3.12: Graph each line, using the given information or equation.m =  14, 42
 3.13: Graph each line, using the given information or equation.m = 0 B
 3.14: Graph each line, using the given information or equation.x + 5y = 0
 3.15: Graph each line, using the given information or equation.y = x + 6
 3.16: Graph each line, using the given information or equation.4x = 3y  24
 3.17: Graph each line, using the given information or equation.x + 4 = 0
 3.18: Graph each line, using the given information or equation.x  3y = 6
 3.19: Match the description in Column I with the correct equation in Colu...
 3.20: Which equations are equivalent to 2x + 5y = 20?A y =  (x  52 25x ...
 3.21: Write an equation for each line. Give the final answer in slopeint...
 3.22: Write an equation for each line. Give the final answer in slopeint...
 3.23: Write an equation for each line. Give the final answer in slopeint...
 3.24: Write an equation for each line. Give the final answer in slopeint...
 3.25: Write an equation for each line. Give the final answer in slopeint...
 3.26: Write an equation for each line. Give the final answer in slopeint...
 3.27: Write an equation for each line. Give the final answer in slopeint...
 3.28: m = 2, b = 4
 3.29: Write an equation for each line. Give the final answer in slopeint...
 3.30: Write an equation for each line. Give the final answer in slopeint...
 3.31: m = through 13, 02
 3.32: Write an equation for each line. Give the final answer in slopeint...
 3.33: Write an equation for each line. Give the final answer in slopeint...
 3.34: Write an equation for each line. Give the final answer in slopeint...
 3.35: Write an equation for each line. Give the final answer in slopeint...
 3.36: Graph each linear inequality3x + 5y 7 9
 3.37: Graph each linear inequality2x  3y 7 6
 3.38: Graph each linear inequalityx  2y 0
 3.39: Decide whether each relation is or is not a function. In Exercises ...
 3.40: Decide whether each relation is or is not a function. In Exercises ...
 3.41: x y
 3.42: y0
 3.43: 2x + 3y = 12
 3.44: y = x2
 3.45: Find (a) 12 and b 1121x2 = 3x + 2
 3.46: Find (a) 12 and b 1121x2 = 2x2  1
 3.47: Find (a) 12 and b 1121x2 =  x + 3 
 3.48: In Exercises 4853, match each statement to the appropriate graph or...
 3.49: In Exercises 4853, match each statement to the appropriate graph or...
 3.50: In Exercises 4853, match each statement to the appropriate graph or...
 3.51: In Exercises 4853, match each statement to the appropriate graph or...
 3.52: In Exercises 4853, match each statement to the appropriate graph or...
 3.53: In Exercises 4853, match each statement to the appropriate graph or...
 3.54: Find the intercepts and the slope of each line. Then graph the line...
 3.55: Find the intercepts and the slope of each line. Then graph the line...
 3.56: Find the intercepts and the slope of each line. Then graph the line...
 3.57: Write an equation for each line. Give the final answer in slopeint...
 3.58: Write an equation for each line. Give the final answer in slopeint...
 3.59: Write an equation for each line. Give the final answer in slopeint...
 3.60: Write an equation for each line. Give the final answer in slopeint...
 3.61: Graph each inequality.x  2y 6
 3.62: Graph each inequality.y 6 4x
 3.63: The percents of fouryear college students in public schools who ea...
 3.64: The percents of fouryear college students in public schools who ea...
 3.65: The percents of fouryear college students in public schools who ea...
 3.66: The percents of fouryear college students in public schools who ea...
Solutions for Chapter 3: Linear Equations and Inequalities in Two Variables; Functions
Full solutions for Beginning Algebra  11th Edition
ISBN: 9780321673480
Solutions for Chapter 3: Linear Equations and Inequalities in Two Variables; Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Beginning Algebra was written by and is associated to the ISBN: 9780321673480. Chapter 3: Linear Equations and Inequalities in Two Variables; Functions includes 66 full stepbystep solutions. Since 66 problems in chapter 3: Linear Equations and Inequalities in Two Variables; Functions have been answered, more than 39967 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11.

Column space C (A) =
space of all combinations of the columns of A.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = 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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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.

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.

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.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).