 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 20009 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.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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.

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

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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

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