 4.1.4.1.1: Fill in each blank so that the resulting statement is true. A solut...
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 4.1.4.1.5: In Exercises 110, determine whether the given ordered pair is a sol...
 4.1.4.1.6: In Exercises 110, determine whether the given ordered pair is a sol...
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 4.1.4.1.14: In Exercises 110, determine whether the given ordered pair is a sol...
 4.1.4.1.15: In Exercises 1142, solve each system by graphing. If there is no so...
 4.1.4.1.16: In Exercises 1142, solve each system by graphing. If there is no so...
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 4.1.4.1.24: In Exercises 1142, solve each system by graphing. If there is no so...
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 4.1.4.1.26: In Exercises 1142, solve each system by graphing. If there is no so...
 4.1.4.1.27: In Exercises 1142, solve each system by graphing. If there is no so...
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 4.1.4.1.29: In Exercises 1142, solve each system by graphing. If there is no so...
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 4.1.4.1.42: In Exercises 1142, solve each system by graphing. If there is no so...
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 4.1.4.1.44: In Exercises 1142, solve each system by graphing. If there is no so...
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 4.1.4.1.47: In Exercises 4350, find the slope and the yintercept for the graph ...
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 4.1.4.1.55: A rental company charges $40.00 a day plus $0.35 per mile to rent a...
 4.1.4.1.56: A band plans to record a demo. Studio A rents for $100 plus $50 per...
 4.1.4.1.57: You plan to start taking an aerobics class. Nonmembers pay $4 per c...
 4.1.4.1.58: What is a system of linear equations? Provide an example with your ...
 4.1.4.1.59: What is a solution of a system of linear equations?
 4.1.4.1.60: Explain how to determine if an ordered pair is a solution of a syst...
 4.1.4.1.61: Explain how to solve a system of linear equations by graphing.
 4.1.4.1.62: What is an inconsistent system? What happens if you attempt to solv...
 4.1.4.1.63: Explain how a linear system can have infinitely many solutions.
 4.1.4.1.64: What are dependent equations? Provide an example with your descript...
 4.1.4.1.65: The following system models the winning times, y, in seconds, in th...
 4.1.4.1.66: In Exercises 6265, determine whether each statement makes sense or ...
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 4.1.4.1.74: Write a system of linear equations whose solution is (5, 1). How ma...
 4.1.4.1.75: Write a system of equations with one solution, a system of equation...
 4.1.4.1.76: Verify your solutions to any five exercises from Exercises 11 throu...
 4.1.4.1.77: Read Exercise 72. Then use a graphing utility to solve the systems ...
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 4.1.4.1.80: Read Exercise 72. Then use a graphing utility to solve the systems ...
 4.1.4.1.81: Read Exercise 72. Then use a graphing utility to solve the systems ...
 4.1.4.1.82: Read Exercise 72. Then use a graphing utility to solve the systems ...
 4.1.4.1.83: Read Exercise 72. Then use a graphing utility to solve the systems ...
 4.1.4.1.84: Read Exercise 72. Then use a graphing utility to solve the systems ...
 4.1.4.1.85: In Exercises 8183, perform the indicated operation 3 (9) (Section 1...
 4.1.4.1.86: In Exercises 8183, perform the indicated operation 3 (9) (Section 1...
 4.1.4.1.87: In Exercises 8183, perform the indicated operation 3(9) (Section 1....
 4.1.4.1.88: Exercises 8486 will help you prepare for the material covered in th...
 4.1.4.1.89: Exercises 8486 will help you prepare for the material covered in th...
 4.1.4.1.90: Exercises 8486 will help you prepare for the material covered in th...
Solutions for Chapter 4.1: Solving Systems of Linear Equations by Graphing
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 4.1: Solving Systems of Linear Equations by Graphing
Get Full SolutionsIntroductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Since 90 problems in chapter 4.1: Solving Systems of Linear Equations by Graphing have been answered, more than 71767 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.1: Solving Systems of Linear Equations by Graphing includes 90 full stepbystep solutions.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

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

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.

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

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

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).

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

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.