 Chapter 3.1: (4, 2) e 2x  5y = 2 3x + 4y = 4
 Chapter 3.2: (5, 3) x + 2y = 11 y =  x 3 + 4 3
 Chapter 3.3: e x + y = 5 3x  y = 3
 Chapter 3.4: e 3x  2y = 6 6x  4y = 12
 Chapter 3.5: y = 3 5 x  3 2x  y = 4
 Chapter 3.6: e y = x + 4 3x + 3y = 6
 Chapter 3.7: e 2x  y = 2 x + 2y = 11
 Chapter 3.8: e y = 2x + 3 3x + 2y = 17
 Chapter 3.9: e 3x + 2y = 8 2x + 5y = 2
 Chapter 3.10: e 5x  2y = 14 3x + 4y = 11
 Chapter 3.11: e y = 4  x 3x + 3y = 12
 Chapter 3.12: x 8 + 3y 4 = 19 8  x 2 + 3y 4 = 1 2
 Chapter 3.13: e x  2y + 3 = 0 2x  4y + 7 = 0
 Chapter 3.14: An appliance store is having a sale on small TVs and stereos. One d...
 Chapter 3.15: You invested $9000 in two funds paying 4% and 7% annual interest. A...
 Chapter 3.16: A chemist needs to mix a solution that is 34% silver nitrate with o...
 Chapter 3.17: When a plane flies with the wind, it can travel 2160 miles in 3 hou...
 Chapter 3.18: The perimeter of a rectangular table top is 34 feet. The difference...
 Chapter 3.19: Find the loss or the gain from selling 400 graphing calculators.
 Chapter 3.20: Determine the breakeven point. Describe what this means.
 Chapter 3.21: Write the profit function, P, from producing and selling x graphing...
 Chapter 3.22: A company is planning to manufacture computer desks. The fixed cost...
 Chapter 3.23: Is (3, 2, 5) a solution of the system x + y + z = 0 2x  3y + z =...
 Chapter 3.24: 2x  y + z = 1 3x  3y + 4z = 5 4x  2y + 3z = 4
 Chapter 3.25: x + 2y  z = 5 2x  y + 3z = 0 2y + z = 1
 Chapter 3.26: 3x  4y + 4z = 7 x  y  2z = 2 2x  3y + 6z = 5
 Chapter 3.27: Find the quadratic function y = ax2 + bx + c whose graph passes thr...
 Chapter 3.28: 20th Century Death The greatest cause of death in the 20th century ...
 Chapter 3.29: J 1 8 0 7 2 3 14 R 1 7 R2
 Chapter 3.30: J 1 3 2 1 2 1 5 R 2R1 + R2
 Chapter 3.31: C 2 2 1 1 2 1 6 4 3 3 1 2 5 S 1 2 R1
 Chapter 3.32: C 1 2 2 0 1 1 0 5 4 3 2 2 1 S 5R2 + R3
 Chapter 3.33: e x + 4y = 7 3x + 5y = 0
 Chapter 3.34: e 2x  3y = 8 6x + 9y = 4
 Chapter 3.35: x + 2y + 3z = 5 2x + y + z = 1 x + y  z = 8
 Chapter 3.36: x  2y + z = 0 y  3z = 1 2y + 5z = 2
 Chapter 3.37: 2 3 2 1 5 2
 Chapter 3.38: 2 3 4 8 2
 Chapter 3.39: 3 2 4 3 1 1 5 2 4 0 3
 Chapter 3.40: 3 4 7 0 5 6 0 3 2 4
 Chapter 3.41: e x  2y = 8 3x + 2y = 1
 Chapter 3.42: e 7x + 2y = 0 2x + y = 3
 Chapter 3.43: x + 2y + 2z = 5 2x + 4y + 7z = 19 2x  5y  2z = 8
 Chapter 3.44: 2x + y = 4 y  2z = 0 3x  2z = 11
 Chapter 3.45: Use the quadratic function y = ax2 + bx + c to model the following ...
Solutions for Chapter Chapter 3: Systems of Linear Equations
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter Chapter 3: Systems of Linear Equations
Get Full SolutionsSince 45 problems in chapter Chapter 3: Systems of Linear Equations have been answered, more than 52326 students have viewed full stepbystep solutions from this chapter. Chapter Chapter 3: Systems of Linear Equations includes 45 full stepbystep solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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).

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

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 .

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

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)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Solvable system Ax = b.
The right side b is in the column space of A.

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