 7.1.1: Can a system of linear equations have exactly two solutions? Explai...
 7.1.2: If you are performing a breakeven analysis for a business and thei...
 7.1.3: If you are solving a breakeven analysis and get a negative breake...
 7.1.4: If you are solving a breakeven analysis and there is no breakeven...
 7.1.5: Given a system of equations, explain at least two different methods...
 7.1.6: For the following exercises, determine whether the given ordered pa...
 7.1.7: For the following exercises, determine whether the given ordered pa...
 7.1.8: For the following exercises, determine whether the given ordered pa...
 7.1.9: For the following exercises, determine whether the given ordered pa...
 7.1.10: For the following exercises, determine whether the given ordered pa...
 7.1.11: For the following exercises, solve each system by substitution. x +...
 7.1.12: For the following exercises, solve each system by substitution. 3x ...
 7.1.13: For the following exercises, solve each system by substitution. 4x ...
 7.1.14: For the following exercises, solve each system by substitution. 2x ...
 7.1.15: For the following exercises, solve each system by substitution. 2x ...
 7.1.16: For the following exercises, solve each system by substitution. x 0...
 7.1.17: For the following exercises, solve each system by substitution. 3 x...
 7.1.18: For the following exercises, solve each system by substitution. 3x ...
 7.1.19: For the following exercises, solve each system by substitution. __ ...
 7.1.20: For the following exercises, solve each system by substitution. 1 _...
 7.1.21: For the following exercises, solve each system by addition. 2x + 5y...
 7.1.22: For the following exercises, solve each system by addition. 6x 5y =...
 7.1.23: For the following exercises, solve each system by addition. 5x y = ...
 7.1.24: For the following exercises, solve each system by addition. 7x 2y =...
 7.1.25: For the following exercises, solve each system by addition. x + 2y ...
 7.1.26: For the following exercises, solve each system by addition. 7x + 6y...
 7.1.27: For the following exercises, solve each system by addition. __ 6 x ...
 7.1.28: For the following exercises, solve each system by addition. 1 __ 3 ...
 7.1.29: For the following exercises, solve each system by addition. 0.2x + ...
 7.1.30: For the following exercises, solve each system by addition. 0.1x + ...
 7.1.31: For the following exercises, solve each system by any method. 5x + ...
 7.1.32: For the following exercises, solve each system by any method. 6x 8y...
 7.1.33: For the following exercises, solve each system by any method. 5x 2y...
 7.1.34: For the following exercises, solve each system by any method. x 5 _...
 7.1.35: For the following exercises, solve each system by any method. . 7x ...
 7.1.36: For the following exercises, solve each system by any method. 3x + ...
 7.1.37: For the following exercises, solve each system by any method. 7 __ ...
 7.1.38: For the following exercises, solve each system by any method. 1 __ ...
 7.1.39: For the following exercises, solve each system by any method. 2.2x ...
 7.1.40: For the following exercises, solve each system by any method. 0.1x ...
 7.1.41: For the following exercises, graph the system of equations and stat...
 7.1.42: For the following exercises, graph the system of equations and stat...
 7.1.43: For the following exercises, graph the system of equations and stat...
 7.1.44: For the following exercises, graph the system of equations and stat...
 7.1.45: For the following exercises, graph the system of equations and stat...
 7.1.46: For the following exercises, use the intersect function on a graphi...
 7.1.47: For the following exercises, use the intersect function on a graphi...
 7.1.48: For the following exercises, use the intersect function on a graphi...
 7.1.49: For the following exercises, use the intersect function on a graphi...
 7.1.50: For the following exercises, use the intersect function on a graphi...
 7.1.51: For the following exercises, solve each system in terms of A, B, C,...
 7.1.52: For the following exercises, solve each system in terms of A, B, C,...
 7.1.53: For the following exercises, solve each system in terms of A, B, C,...
 7.1.54: For the following exercises, solve each system in terms of A, B, C,...
 7.1.55: For the following exercises, solve each system in terms of A, B, C,...
 7.1.56: For the following exercises, solve for the desired quantity. A stuf...
 7.1.57: For the following exercises, solve for the desired quantity. A fast...
 7.1.58: For the following exercises, solve for the desired quantity. A cell...
 7.1.59: For the following exercises, solve for the desired quantity. A musi...
 7.1.60: For the following exercises, solve for the desired quantity. A guit...
 7.1.61: For the following exercises, use a system of linear equations with ...
 7.1.62: For the following exercises, use a system of linear equations with ...
 7.1.63: For the following exercises, use a system of linear equations with ...
 7.1.64: For the following exercises, use a system of linear equations with ...
 7.1.65: For the following exercises, use a system of linear equations with ...
 7.1.66: For the following exercises, use a system of linear equations with ...
 7.1.67: For the following exercises, use a system of linear equations with ...
 7.1.68: For the following exercises, use a system of linear equations with ...
 7.1.69: For the following exercises, use a system of linear equations with ...
 7.1.70: For the following exercises, use a system of linear equations with ...
 7.1.71: For the following exercises, use a system of linear equations with ...
 7.1.72: For the following exercises, use a system of linear equations with ...
 7.1.73: For the following exercises, use a system of linear equations with ...
 7.1.74: For the following exercises, use a system of linear equations with ...
 7.1.75: For the following exercises, use a system of linear equations with ...
 7.1.76: For the following exercises, use a system of linear equations with ...
 7.1.77: For the following exercises, use a system of linear equations with ...
Solutions for Chapter 7.1: SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES
Full solutions for College Algebra  1st Edition
ISBN: 9781938168383
Solutions for Chapter 7.1: SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES
Get Full SolutionsSince 77 problems in chapter 7.1: SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES have been answered, more than 30258 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 1. Chapter 7.1: SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES includes 77 full stepbystep solutions. College Algebra was written by and is associated to the ISBN: 9781938168383.

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.)

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).

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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!).

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

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

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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 •

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.

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.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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