 4.5.1: A method for finding the maximum or minimum value of a quantity tha...
 4.5.2: An algebraic expression in two or more variables describing a quant...
 4.5.3: A system of linear inequalities is used to represent restrictions, ...
 4.5.4: Objective Function z = 30x + 45y
 4.5.5: Objective Function z = 3x + 2y Constraints
 4.5.6: Objective Function z = 2x + 3y Constraints x 0, y 0 2x + y 8 2x + 3...
 4.5.7: Objective Function z = 4x + y Constraints x 0, y 0 2x + 3y 12 x + y 3
 4.5.8: Objective Function z = x + 6y Constraints x 0, y 0 2x + y 10 x  2y...
 4.5.9: Objective Function z = 3x  2y Constraints 1 x 5 y 2 x  y 3
 4.5.10: Objective Function z = 5x  2y Constraints 0 x 5 0 y 3 x + y 2
 4.5.11: Objective Function z = 4x + 2y
 4.5.12: Objective Function z = 2x + 4y Constraints x 0, y 0 x + 3y 6 x + y ...
 4.5.13: Objective Function z = 10x + 12y Constraints x 0, y 0 2x + y 10 2x ...
 4.5.14: Objective Function z = 5x + 6y Constraints x 0, y 0 2x + y 10 x + 2...
 4.5.15: A television manufacturer makes rearprojection and plasma televisi...
 4.5.16: a. A student earns $10 per hour for tutoring and $7 per hour as a t...
 4.5.17: A manufacturer produces two models of mountain bicycles. The times ...
 4.5.18: A large institution is preparing lunch menus containing foods A and...
 4.5.19: Food and clothing are shipped to survivors of a hurricane. Each car...
 4.5.20: On June 24, 1948, the former Soviet Union blocked all land and wate...
 4.5.21: A theater is presenting a program on drinking and driving for stude...
 4.5.22: You are about to take a test that contains computation problems wor...
 4.5.23: In 1978, a ruling by the Civil Aeronautics Board allowed Federal Ex...
 4.5.24: What kinds of problems are solved using the linear programming method?
 4.5.25: What is an objective function in a linear programming problem?
 4.5.26: What is a constraint in a linear programming problem? How is a cons...
 4.5.27: In your own words, describe how to solve a linear programming problem.
 4.5.28: Describe a situation in your life in which you would like to maximi...
 4.5.29: In order to solve a linear programming problem, I use the graph rep...
 4.5.30: I use the coordinates of each vertex from my graph representing the...
 4.5.31: I need to be able to graph systems of linear inequalities in order ...
 4.5.32: An important application of linear programming for businesses invol...
 4.5.33: Suppose that you inherit $10,000. The will states how you must inve...
 4.5.34: Consider the objective function z = Ax + By (A 7 0 and B 7 0) subje...
 4.5.35: Simplify: (2x4y3 )(3xy4 ) 3 . (Section 1.6, Example 9)
 4.5.36: Solve for L: 3P = 2L  W 4 . (Section 1.5, Example 8)
 4.5.37: If f(x) = x3 + 2x2  5x + 4, find f(1). (Section 2.1, Example 3)
 4.5.38: (9x3 + 7x2  5x + 3) + (13x3 + 2x2  8x  6)
 4.5.39: (7x3  8x2 + 9x  6)  (2x3  6x2  3x + 9)
 4.5.40: The figures show the graphs of two functions. x y 10 20 10 20 4 2 2...
Solutions for Chapter 4.5: Linear Programming
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 4.5: Linear Programming
Get Full SolutionsThis textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Since 40 problems in chapter 4.5: Linear Programming have been answered, more than 19877 students have viewed full stepbystep solutions from this chapter. Chapter 4.5: Linear Programming includes 40 full stepbystep solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934.

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.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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 .

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 •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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).

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).

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 ().

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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