 9.1: x y 2 x y 0
 9.2: 2x x 3y y 3 0
 9.3: 4x 8x y y 1 17 0 0
 9.4: 10x x 6y 9y 14 7 0 0
 9.5: 0.5x 1.25x y 4.5y 0.75 2.5
 9.6: x x 2 5y 1 5y 3 5 4 5
 9.7: x2 y2 9 x y 1
 9.8: x2 y2 169 3x 2y 39
 9.9: y 2x2 y x 4 2x2
 9.10: x y 3 x y2 1
 9.11: 2x y x 5y 10 6
 9.12: 8x 2x 3y 5y 3 28
 9.13: y y 2x2 4x 1 x2 4x 3
 9.14: y2 2y x 0 x y 0
 9.15: 2ex y y 2ex 0
 9.16: x2 2x y2 3y 100 12 2
 9.17: y 2 log x y 3 4x 5
 9.18: y y ln x 1 3 4 1 2 x
 9.19: BREAKEVEN ANALYSIS You set up a scrapbook business and make an ini...
 9.20: CHOICE OF TWO JOBS You are offered two sales jobs at a pharmaceutic...
 9.21: GEOMETRY The perimeter of a rectangle is 480 meters and its length ...
 9.22: GEOMETRY The perimeter of a rectangle is 68 feet and its width is t...
 9.23: GEOMETRY The perimeter of a rectangle is 40 inches. The area of the...
 9.24: BODY MASS INDEX Body Mass Index (BMI) is a measure of body fat base...
 9.25: 2x y 2 6x 8y 39 9
 9.26: 40x 30y 20x 50y 24 14
 9.27: 0.2x 0.3y 0.14 0.4x 0.5y 0.20
 9.28: 12x 42y 30x 18y 17 19
 9.29: 3x 2y 0 3x 2 y 5 10
 9.30: 7x 12y 63 2x 3 y 2 21
 9.31: 1.25x 5x 2y 8y 3.5 14
 9.32: 1.5x 6x 2.5y 10y 8.5 24
 9.33: x 5y 4 x 3y 6
 9.34: 3x y 7 9x 3y 21
 9.35: 3x y 7 6x 2y 8
 9.36: 2x y 3 x 5y 4
 9.37: p 37 0.0002x p
 9.38: p 120 0.0001x p 45 0.0002x
 9.39: x 4y 3z y z z 3 1 5 9.3
 9.40: x 7y 8z y 9z z 85 35 3
 9.41: 4x 3y 8y 2z 7z z 65 14 10 x
 9.42: 5x 3y 7z 8z z 9 4 7 4
 9.43: x 3x 4x 2y 2y 6z z 2z 4 4 16
 9.44: x 2x 4x 3y y z 13 5z 23 2z 14 x
 9.45: x 2y 2x 3y x 3y z 3z 6 7 11 x
 9.46: 2x 3x 3x 2y y 6z 9 11z 16 7z 11 x
 9.47: x 4x 3y 2y y z 2z 4w w 3w 1 4 2 5
 9.48: x 3x 2x x y 4y 3y 4y z z z w w 3w 2w 6 3 6 7
 9.49: 5x 12y 7z 16 3x 7y 4z 9
 9.50: 2x 5y 19z 34 3x 8y 31z 54
 9.51: 4 4 4 (1, 2) (0, 5) (2, 5) y
 9.52: 1, 0) (5, 6) 12 6 6 12 24 x y
 9.53: 1 4 32 1 5 (2, 1) (5, 2) (1, 2) x y
 9.54: ) 6 2 24 2 8 x y
 9.55: DATA ANALYSIS: ONLINE SHOPPING The table shows the projected online...
 9.56: AGRICULTURE A mixture of 6 gallons of chemical A, 8 gallons of chem...
 9.57: INVESTMENT ANALYSIS An inheritance of $40,000 was divided among thr...
 9.58: VERTICAL MOTION An object moving vertically is at the given heights...
 9.59: SPORTS Pebble Beach Golf Links in Pebble Beach, California is an 18...
 9.60: SPORTS St Andrews Golf Course in St Andrews, Scotland is one of the...
 9.61: 3 x2 20x
 9.62: x 8 x2 3x 28 3
 9.63: 3x 4 x3 5x2 x
 9.64: x 2 x x2 22
 9.65: 4 x x2 6x 8
 9.66: x x2 3x 2
 9.67: x2 x2 2x 15
 9.68: 9 x2 9
 9.69: x2 2x x3 x2 x 1 9
 9.70: 4x 3 x 12
 9.71: 3x2 4x x2 12
 9.72: 4x2 x 1 x2 1
 9.73: y 5 3y x 7 1 2 x 9
 9.74: 3y x 7 1
 9.75: y 4x2 > 1 y
 9.76: y 3 x2 2
 9.77: x 12 y 32 < 16 y
 9.78: x2 y 52 > 1
 9.79: x 3x 2y 160 y 180 x 0 y 0
 9.80: 2x 3y 24 2x y 16 x 0 y 0
 9.81: 3x 2y 24 x 2y 12 2 x 15 y 15
 9.82: 2x y 16 x 3y 18 0 x 25 0 y 25
 9.83: y < x 1 y > x2 1
 9.84: y 6 2x x2 y x 6
 9.85: 2x 3y 0 2x y 8 y 0
 9.86: x2 y2 9 x 32 y2 9
 9.87: INVENTORY COSTS A warehouse operator has 24,000 square feet of floo...
 9.88: NUTRITION A dietitian is asked to design a special dietary suppleme...
 9.89: p 160 0.0001x p 70 0.0002x
 9.90: p 130 0.0002x p 30 0.0003x
 9.91: GEOMETRY Derive a set of inequalities to describe the region of a r...
 9.92: DATA ANALYSIS: COMPUTER SALES The table shows the sales (in billion...
 9.93: Objective function:
 9.94: Objective function: Constraints: Constraints:
 9.95: Objective function
 9.96: Objective function:
 9.97: Objective function:
 9.98: Objective function:
 9.99: OPTIMAL REVENUE A student is working part time as a hairdresser to ...
 9.100: OPTIMAL PROFIT A shoe manufacturer produces a walking shoe and a ru...
 9.101: OPTIMAL PROFIT A manufacturer produces two models of bicycles. The ...
 9.102: OPTIMAL COST A pet supply company mixes two brands of dry dog food....
 9.103: OPTIMAL COST Regular unleaded gasoline and premium unleaded gasolin...
 9.104: If a system of equations consists of a circle and a parabola, it is...
 9.105: The system represents the region covered by an isosceles trapezoid.
 9.106: It is possible for an objective function of a linear programming pr...
 9.107: 8, 10
 9.108: 5, 4
 9.109: 4 3, 2003
 9.110: 2, 11 5 4
 9.111: 4, 1, 2003
 9.112: 3, 5, 2006
 9.113: 5, 3 2, 2
 9.114: 1 2, 2, 3 4
 9.115: WRITING Explain what is meant by an inconsistent system of linear e...
 9.116: How can you tell graphically that a system of linear equations in t...
Solutions for Chapter 9: Systems of Equations and Inequlities
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 9: Systems of Equations and Inequlities
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. Since 116 problems in chapter 9: Systems of Equations and Inequlities have been answered, more than 48888 students have viewed full stepbystep solutions from this chapter. Chapter 9: Systems of Equations and Inequlities includes 116 full stepbystep solutions.

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

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

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.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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!).

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.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as 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!

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.