 1.1: In Exercises 1 to 20, solve each equation.
 1.2: In Exercises 1 to 20, solve each equation.
 1.3: In Exercises 1 to 20, solve each equation.
 1.4: In Exercises 1 to 20, solve each equation.
 1.5: In Exercises 1 to 20, solve each equation.
 1.6: In Exercises 1 to 20, solve each equation.
 1.7: In Exercises 1 to 20, solve each equation.
 1.8: In Exercises 1 to 20, solve each equation.
 1.9: In Exercises 1 to 20, solve each equation.
 1.10: In Exercises 1 to 20, solve each equation.
 1.11: In Exercises 1 to 20, solve each equation.
 1.12: In Exercises 1 to 20, solve each equation.
 1.13: In Exercises 1 to 20, solve each equation.
 1.14: In Exercises 1 to 20, solve each equation.
 1.15: In Exercises 1 to 20, solve each equation.
 1.16: In Exercises 1 to 20, solve each equation.
 1.17: In Exercises 1 to 20, solve each equation.
 1.18: In Exercises 1 to 20, solve each equation.
 1.19: In Exercises 1 to 20, solve each equation.
 1.20: In Exercises 1 to 20, solve each equation.
 1.21: In Exercises 21 and 22, use the discriminant to determine whether t...
 1.22: In Exercises 21 and 22, use the discriminant to determine whether t...
 1.23: In Exercises 23 to 40, solve each equation.
 1.24: In Exercises 23 to 40, solve each equation.
 1.25: In Exercises 23 to 40, solve each equation.
 1.26: In Exercises 23 to 40, solve each equation.
 1.27: In Exercises 23 to 40, solve each equation.
 1.28: In Exercises 23 to 40, solve each equation.
 1.29: In Exercises 23 to 40, solve each equation.
 1.30: In Exercises 23 to 40, solve each equation.
 1.31: In Exercises 23 to 40, solve each equation.
 1.32: In Exercises 23 to 40, solve each equation.
 1.33: In Exercises 23 to 40, solve each equation.
 1.34: In Exercises 23 to 40, solve each equation.
 1.35: In Exercises 23 to 40, solve each equation.
 1.36: In Exercises 23 to 40, solve each equation.
 1.37: In Exercises 23 to 40, solve each equation.
 1.38: In Exercises 23 to 40, solve each equation.
 1.39: In Exercises 23 to 40, solve each equation.
 1.40: In Exercises 23 to 40, solve each equation.
 1.41: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.42: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.43: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.44: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.45: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.46: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.47: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.48: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.49: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.50: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.51: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.52: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.53: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.54: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.55: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.56: In Exercises 41 to 56, solve each inequality. Write the answer usin...
 1.57: Rectangular Region The length of a rectangle is 9 feet less than tw...
 1.58: Rectangular Region The perimeter of a rectangle is 40 inches and it...
 1.59: Height of a Tree The height of a tree is estimated by using its sha...
 1.60: Shadow Length A person 5 feet 6 inches tall is walking away from a ...
 1.61: Diameter of a Cone As sand is poured from a chute, it forms a right...
 1.62: Individual Price A calculator and a battery together sell for $21. ...
 1.63: Maintenance Cost Eighteen owners share the maintenance cost of a co...
 1.64: Investment A total of $5500 was deposited into two simple interest ...
 1.65: Distance to an Island A motorboat left a harbor and traveled to an ...
 1.66: Running Inez can run at a rate that is 2 miles per hour faster than...
 1.67: Chemistry A chemist mixes a 5% salt solution with an 11% salt solut...
 1.68: Pharmacy How many milliliters of pure water should a pharmacist add...
 1.69: Alloys How many ounces of a gold alloy that costs $460 per ounce mu...
 1.70: Blends A grocer makes a snack mixture of raisins and nuts by combin...
 1.71: Construction of a Wall A mason can build a wall in 9 hours less tha...
 1.72: Parallel Processing One computer can solve a problem 5 minutes fast...
 1.73: Dogs on a Beach Two dogs start, at the same time, from points C and...
 1.74: Constructing a Box A square piece of cardboard is formed into a box...
 1.75: Sports In an Olympic 10meter diving competition, the height h, in ...
 1.76: Fair Coin If a fair coin is tossed 100 times, we would expect heads...
 1.77: Mean Height If a researcher wanted to know the mean height (the mea...
 1.78: Mean Waist Size If a researcher wanted to know the mean waist size ...
 1.79: Basketball Dimensions A basketball is to have a circumference of 29...
 1.80: Population Density The population density in people per square mile...
 1.81: Physics Force F is directly proportional to acceleration a. If a fo...
 1.82: Physics The distance an object will fall on the moon is directly pr...
 1.83: Business The number of MP3 players a company can sell is inversely ...
 1.84: Magnetism The repulsive force between the north poles of two magnet...
 1.85: Acceleration The acceleration due to gravity on the surface of a pl...
Solutions for Chapter 1: College Algebra and Trigonometry 7th Edition
Full solutions for College Algebra and Trigonometry  7th Edition
ISBN: 9781439048603
Solutions for Chapter 1
Get Full SolutionsSince 85 problems in chapter 1 have been answered, more than 12380 students have viewed full stepbystep solutions from this chapter. Chapter 1 includes 85 full stepbystep solutions. College Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7.

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.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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!).

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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