 4.4.4.1.254: Fill in each blank so that the resulting statement is true. Suppose...
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 4.4.4.1.261: In Exercises 14, let x represent one number and let y represent the...
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 4.4.4.1.268: (In Exercises 810 refer to the bar graph to visualize that your sol...
 4.4.4.1.269: (In Exercises 810 refer to the bar graph to visualize that your sol...
 4.4.4.1.270: (In Exercises 810 refer to the bar graph to visualize that your sol...
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 4.4.4.1.272: A restaurant purchased eight tablecloths and five napkins for $106....
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 4.4.4.1.279: You are choosing between two plans at a discount warehouse. Plan A ...
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 4.4.4.1.287: In Exercises 2728, an isosceles triangle in which angles B and C ha...
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 4.4.4.1.289: Exercises 2934 involve dual investments You invested $7000 in two a...
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 4.4.4.1.293: Exercises 2934 involve dual investments Things did not go quite as ...
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 4.4.4.1.310: Use Figure 4.11 on page 307 to write a word problem similar to Exam...
 4.4.4.1.311: Exercises 1720 involve using systems of linear equations to compare...
 4.4.4.1.312: Describe two similarities between interest and mixture problems.
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 4.4.4.1.314: In Exercises 5457, determine whether each statement makes sense or ...
 4.4.4.1.315: In Exercises 5457, determine whether each statement makes sense or ...
 4.4.4.1.316: In Exercises 5457, determine whether each statement makes sense or ...
 4.4.4.1.317: In Exercises 5457, determine whether each statement makes sense or ...
 4.4.4.1.318: A set of identical twins can only be distinguished by the character...
 4.4.4.1.319: Tourist: How many birds and lions do you have in your zoo? Zookeepe...
 4.4.4.1.320: Find the measure of each angle whose degree measure is represented ...
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 4.4.4.1.323: You have $70,000 to invest. Part of the money is to be placed in a ...
 4.4.4.1.324: Select any two problems that you solved from Exercises 526. Use a g...
 4.4.4.1.325: Solve: 2(x 3) 24 2(x 4). (Section 2.3, Example 3)
 4.4.4.1.326: Simplify: 5 6(x 1). (Section 1.7, Example 7)
 4.4.4.1.327: Write the slopeintercept form of the equation of the line passing ...
 4.4.4.1.328: Exercises 6870 will help you prepare for the material covered in th...
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 4.4.4.1.330: Exercises 6870 will help you prepare for the material covered in th...
Solutions for Chapter 4.4: Problem Solving Using Systems of Equations
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 4.4: Problem Solving Using Systems of Equations
Get Full SolutionsIntroductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. Chapter 4.4: Problem Solving Using Systems of Equations includes 77 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Since 77 problems in chapter 4.4: Problem Solving Using Systems of Equations have been answered, more than 68268 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Column space C (A) =
space of all combinations of the columns of A.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Outer product uv T
= column times row = rank one matrix.

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.

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 ().

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

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.

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