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 5.1.2: In Exercises 14, determine whether the gh1tm ordered pair is a sol...
 5.1.3: In Exercises 14, determine whether the gh1tm ordered pair is a sol...
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 5.1.51: For the linear runction f(x)  111x + b,f( 2)  11 and !(3)   9....
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 5.1.53: Use the graphs of the linear functions to solve Exercises 5354. Wr...
 5.1.54: Use the graphs of the linear functions to solve Exercises 5354. Wr...
 5.1.55: The figure shows the graphs of the cost and revemre fimclions for a...
 5.1.56: The figure shows the graphs of the cost and revemre fimclions for a...
 5.1.57: The figure shows the graphs of the cost and revemre fimclions for a...
 5.1.58: The figure shows the graphs of the cost and revemre fimclions for a...
 5.1.59: The figure shows the graphs of the cost and revemre fimclions for a...
 5.1.60: The figure shows the graphs of the cost and revemre fimclions for a...
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 5.1.65: An important appliroJiotl of syslerns of equations mi'ies in connli...
 5.1.66: An important appliroJiotl of syslerns of equations mi'ies in connli...
 5.1.67: The bar graph indic.ates that fewer U.S. adults are getting married...
 5.1.68: The bar graph shows the percentage of Americans for and against the...
 5.1.69: We opened this section with a study showing that late in the semest...
 5.1.70: Although Social Security is a problem, some projections indicate th...
 5.1.71: The graphs show the percentage of American adults in two living arr...
 5.1.72: The graphs show that the U.S. public has taken a more conservative ...
 5.1.73: Use a system of linear equations to solve Exi!rdses 7384. Looking ...
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 5.1.85: What is a system of linear equations? Provide an example with your ...
 5.1.86: What is the solution of a system of linear equations?
 5.1.87: Explain how to solve a system o f equations using the substitution ...
 5.1.88: Explain how to solve a system of equations using the addition metho...
 5.1.89: When is it easier to use the addition method rather than the substi...
 5.1.90: When using the addition or substitution method, how can you tell if...
 5.1.91: When using the addition or substitution method, how can you tell if...
 5.1.92: Describe the breakeven point for a business.
 5.1.93: Verify your solutions to any five exercises in Exercises 542 by us...
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 5.1.98: Write a system of equations having (( 2. 7)1 as a solution set. (M...
 5.1.99: Solve the system for x a nd y in terms of a1.bt. c1 ,a1,b2, and c2:...
 5.1.100: Two identical twins can only be distinguished by the characteristic...
 5.1.101: A marching band has 52 members, and there are 24 in the pompom squ...
 5.1.102: The group should write four different word problems that can be sol...
 5.1.103: Exercises 103105 will help you prepare for lire material covered n...
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Solutions for Chapter 5.1: Systems of Linear Equations in Two Variables
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter 5.1: Systems of Linear Equations in Two Variables
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra , edition: 6. Chapter 5.1: Systems of Linear Equations in Two Variables includes 105 full stepbystep solutions. Since 105 problems in chapter 5.1: Systems of Linear Equations in Two Variables have been answered, more than 39604 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321782281. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.