 2.2.2.2.1: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.2: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.3: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.4: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.5: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.6: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.7: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.8: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.9: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.10: Write the following as algebraic expressions. Then simplify. See Ex...
 2.2.2.2.11: Solve. For Exercises 11 and 12, the solutions have been started for...
 2.2.2.2.12: Solve. For Exercises 11 and 12, the solutions have been started for...
 2.2.2.2.13: Start the solution:1. UNDERSTAND the problem. Reread it as many tim...
 2.2.2.2.14: Start the solution:1. UNDERSTAND the problem. Reread it as many tim...
 2.2.2.2.15: Start the solution:1. UNDERSTAND the problem. Reread it as many tim...
 2.2.2.2.16: Start the solution:1. UNDERSTAND the problem. Reread it as many tim...
 2.2.2.2.17: Start the solution:1. UNDERSTAND the problem. Reread it as many tim...
 2.2.2.2.18: Start the solution:1. UNDERSTAND the problem. Reread it as many tim...
 2.2.2.2.19: Start the solution:1. UNDERSTAND the problem. Reread it as many tim...
 2.2.2.2.20: Start the solution:1. UNDERSTAND the problem. Reread it as many tim...
 2.2.2.2.21: The following graph is called a circle graph or a pie chart. The ci...
 2.2.2.2.22: The following graph is called a circle graph or a pie chart. The ci...
 2.2.2.2.23: The following graph is called a circle graph or a pie chart. The ci...
 2.2.2.2.24: The following graph is called a circle graph or a pie chart. The ci...
 2.2.2.2.25: Use the diagrams to find the unknown measures of angles or lengths ...
 2.2.2.2.26: Use the diagrams to find the unknown measures of angles or lengths ...
 2.2.2.2.27: Use the diagrams to find the unknown measures of angles or lengths ...
 2.2.2.2.28: Use the diagrams to find the unknown measures of angles or lengths ...
 2.2.2.2.29: Use the diagrams to find the unknown measures of angles or lengths ...
 2.2.2.2.30: Use the diagrams to find the unknown measures of angles or lengths ...
 2.2.2.2.31: Solve. See Example 6.The sum of three consecutive integers is 228. ...
 2.2.2.2.32: Solve. See Example 6.The sum of three consecutive odd integers is 3...
 2.2.2.2.33: Solve. See Example 6.The ZIP codes of three Nevada locationsFallon,...
 2.2.2.2.34: Solve. See Example 6.During a recent year, the average SAT scores i...
 2.2.2.2.35: Solve. See Examples 1 through 6.Many companies predict the growth o...
 2.2.2.2.36: Solve. See Examples 1 through 6.Many companies predict the growth o...
 2.2.2.2.37: Solve.The occupations of biomedical engineers, skin care specialist...
 2.2.2.2.38: Solve.The occupations of farmer or rancher, file clerk, and telemar...
 2.2.2.2.39: Solve.The B767300ER aircraft has 88 more seats than the B737 200 ...
 2.2.2.2.40: Solve.Cowboy Stadium, home of the Dallas Cowboys of the NFL, seats ...
 2.2.2.2.41: Solve.A new fax machine was recently purchased for an office in Hop...
 2.2.2.2.42: Solve.A premedical student at a local university was complaining th...
 2.2.2.2.43: Solve.The median compensation for a U.S. university president was $...
 2.2.2.2.44: Solve.In 2009, the population of Brazil was 191.5 million. This rep...
 2.2.2.2.45: Solve.In 2010, the population of Swaziland was 1,200,000 people. Fr...
 2.2.2.2.46: Solve.Dana, an auto parts supplier headquartered in Toledo, Ohio, r...
 2.2.2.2.47: Recall that two angles are complements of each other if their sum i...
 2.2.2.2.48: Recall that two angles are complements of each other if their sum i...
 2.2.2.2.49: Recall that the sum of the angle measures of a triangle is 180.Find...
 2.2.2.2.50: Recall that the sum of the angle measures of a triangle is 180.Find...
 2.2.2.2.51: Recall that the sum of the angle measures of a triangle is 180.Two ...
 2.2.2.2.52: Recall that the sum of the angle measures of a triangle is 180.Two ...
 2.2.2.2.53: The sum of the first and third of three consecutive even integers i...
 2.2.2.2.54: The sum of the second and fourth of four consecutive integers is 11...
 2.2.2.2.55: Daytona International Speedway in Florida has 37,000 more grandstan...
 2.2.2.2.56: For the 20102011 National Hockey League season, the payroll for the...
 2.2.2.2.57: The sum of the populations of the metropolitan regions of New York,...
 2.2.2.2.58: The airports in London, Paris, and Frankfurt have a total of 177.1 ...
 2.2.2.2.59: Suppose the perimeter of the triangle in Example 1b in this section...
 2.2.2.2.60: Suppose the perimeter of the trapezoid in Practice 1b in this secti...
 2.2.2.2.61: Incandescent, fluorescent, and halogen bulbs are lasting longer tod...
 2.2.2.2.62: Falkland Islands, Iceland, and Norway are the top three countries t...
 2.2.2.2.63: During the 2010 Major League Baseball season, the number of wins fo...
 2.2.2.2.64: In the 2010 Winter Olympics, Austria won more medals than the Russi...
 2.2.2.2.65: The three tallest hospitals in the world are Guys Tower in London, ...
 2.2.2.2.66: The official manual for traffic signs is the Manual on Uniform Traf...
 2.2.2.2.67: Find the value of each expression for the given values. See Section...
 2.2.2.2.68: Find the value of each expression for the given values. See Section...
 2.2.2.2.69: Find the value of each expression for the given values. See Section...
 2.2.2.2.70: Find the value of each expression for the given values. See Section...
 2.2.2.2.71: Find the value of each expression for the given values. See Section...
 2.2.2.2.72: Find the value of each expression for the given values. See Section...
 2.2.2.2.73: For Exercise 36, the percents have a sum of 300%. Is this possible?...
 2.2.2.2.74: In your own words, explain the differences in the tables for Exerci...
 2.2.2.2.75: Find an angle such that its supplement is equal to 10 times its com...
 2.2.2.2.76: Find an angle such that its supplement is equal to twice its comple...
 2.2.2.2.77: The average annual number of cigarettes smoked by an American adult...
 2.2.2.2.78: The average annual number of cigarettes smoked by an American adult...
 2.2.2.2.79: The average annual number of cigarettes smoked by an American adult...
 2.2.2.2.80: The average annual number of cigarettes smoked by an American adult...
 2.2.2.2.81: The average annual number of cigarettes smoked by an American adult...
 2.2.2.2.82: The average annual number of cigarettes smoked by an American adult...
 2.2.2.2.83: To break even in a manufacturing business, income or revenue R must...
 2.2.2.2.84: To break even in a manufacturing business, income or revenue R must...
 2.2.2.2.85: To break even in a manufacturing business, income or revenue R must...
 2.2.2.2.86: To break even in a manufacturing business, income or revenue R must...
Solutions for Chapter 2.2: An Introduction to Problem Solving
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 2.2: An Introduction to Problem Solving
Get Full SolutionsIntermediate Algebra was written by and is associated to the ISBN: 9780321785046. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.2: An Introduction to Problem Solving includes 86 full stepbystep solutions. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Since 86 problems in chapter 2.2: An Introduction to Problem Solving have been answered, more than 62471 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

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).

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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 ().

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

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.

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