 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 5920 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.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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

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

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

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).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

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).

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

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