 5.2.1: In Exercises 14, determine if tire given ordered triple is a solut...
 5.2.2: In Exercises 14, determine if tire given ordered triple is a solut...
 5.2.3: In Exercises 14, determine if tire given ordered triple is a solut...
 5.2.4: In Exercises 14, determine if tire given ordered triple is a solut...
 5.2.5: Solve each system in Exercises 518. x + y + 2z  11x + y + 3z  14...
 5.2.6: Solve each system in Exercises 518. ~t + /  2z  ~ .>X  .>J  Z ...
 5.2.7: Solve each system in Exercises 518. 4x  y +2z  11 { 7. .t + 2y ...
 5.2.8: Solve each system in Exercises 518. x  y +3z  8 3x + y  2z   ...
 5.2.9: Solve each system in Exercises 518. 3.t + 2y  3z   2 { 9. 2t  ...
 5.2.10: Solve each system in Exercises 518. 2t + 3y + 7z  13 3x + 2}'  5...
 5.2.11: Solve each system in Exercises 518. .t  4y + 3z  17 {.t+ z  3x+...
 5.2.12: Solve each system in Exercises 518. t+ z  312. x+2y  z  1 2r  ...
 5.2.13: Solve each system in Exercises 518. r.t+ y  2 { x+ y  z  4 3x+2...
 5.2.14: Solve each system in Exercises 518. x + 3y + 5z  20 y  4z 16 3...
 5.2.15: Solve each system in Exercises 518. X+ y   412t + y + 3z   21
 5.2.16: Solve each system in Exercises 518. x+ y  4t + z  4y +z  4
 5.2.17: Solve each system in Exercises 518. 3(2t + y) + 5z   I2(.r  3y ...
 5.2.18: Solve each system in Exercises 518. h  3  2(x  3y)5y + 3z  7 ...
 5.2.19: In Exercises 1922, find the quadratic function y  ax1 + bx + c wh...
 5.2.20: In Exercises 1922, find the quadratic function y  ax1 + bx + c wh...
 5.2.21: In Exercises 1922, find the quadratic function y  ax1 + bx + c wh...
 5.2.22: In Exercises 1922, find the quadratic function y  ax1 + bx + c wh...
 5.2.23: In Exercises 2324, let x represent the first number, y the second ...
 5.2.24: In Exercises 2324, let x represent the first number, y the second ...
 5.2.25: Solve each system in Exercises 2526. x + 2 y +4 z  +   06 3...
 5.2.26: Solve each system in Exercises 2526. x+ 3 y  1 z + 2 32 2 +...
 5.2.27: In Exercises 27 28, find 11. equation of the quadratic function y...
 5.2.28: In Exercises 27 28, find 11. equation of the quadratic function y...
 5.2.29: In Exercises 2930, solve each system for (x, y, .:) in terms of th...
 5.2.30: In Exercises 2930, solve each system for (x, y, .:) in terms of th...
 5.2.31: You 1hrow a ball S\!"ight up from a rooftop. The ball misses the ro...
 5.2.32: A mathematical model can be used to describe the relationship betwe...
 5.2.33: Use a system of linear equations in three variables to solve Exerci...
 5.2.34: Use a system of linear equations in three variables to solve Exerci...
 5.2.35: Use a system of linear equations in three variables to solve Exerci...
 5.2.36: Use a system of linear equations in three variables to solve Exerci...
 5.2.37: Use a system of linear equations in three variables to solve Exerci...
 5.2.38: Use a system of linear equations in three variables to solve Exerci...
 5.2.39: Use a system of linear equations in three variables to solve Exerci...
 5.2.40: Use a system of linear equations in three variables to solve Exerci...
 5.2.41: In the following triangle, the degree measure.s of the three interi...
 5.2.42: What is a system of linear equations in three variables?
 5.2.43: How do you determine whether a given ordered triple is a solution o...
 5.2.44: Describe in general terms how to solve a system in three variables.
 5.2.45: AIDS is taking a deadly toll on southern Africa. Describe how to us...
 5.2.46: Does your graphing utility have a feature that allows you to solve ...
 5.2.47: Verify your results in Exercises 1922 by using a graphing utility ...
 5.2.48: In Exercises 4851, determine whether each statem~nt makes sense or...
 5.2.49: In Exercises 4851, determine whether each statem~nt makes sense or...
 5.2.50: In Exercises 4851, determine whether each statem~nt makes sense or...
 5.2.51: In Exercises 4851, determine whether each statem~nt makes sense or...
 5.2.52: Describe how the system l x + y  z  2w   8 x  2y + 3z + w  18...
 5.2.53: A modernistic painting consists of triangles, rectangles, and penta...
 5.2.54: Group members should develop appropriate functions that model each ...
 5.2.55: Exercises 5557 will help you prepare for the material covered in t...
 5.2.56: Exercises 5557 will help you prepare for the material covered in t...
 5.2.57: Exercises 5557 will help you prepare for the material covered in t...
Solutions for Chapter 5.2: Systems of Linear Equations in Three Variables
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter 5.2: Systems of Linear Equations in Three Variables
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra , edition: 6. Since 57 problems in chapter 5.2: Systems of Linear Equations in Three Variables have been answered, more than 37286 students have viewed full stepbystep solutions from this chapter. Chapter 5.2: Systems of Linear Equations in Three Variables includes 57 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780321782281.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det 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).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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