 1.6.1: The equation is a ________ equation in written in general form.
 1.6.2: Squaring each side of an equation, multiplying each side of an equa...
 1.6.3: The equation is of ________ ________.
 1.6.4: To clear the equation of fractions, multiply each side of the equat...
 1.6.5: To clear the equation of fractions, multiply each side of the equat...
 1.6.6:
 1.6.7:
 1.6.8:
 1.6.9:
 1.6.10:
 1.6.11: 5x
 1.6.12: 9x4 24x3 16x 5x 2 0
 1.6.13: x3 3x 2 x 3 0 9
 1.6.14: x3 2x 2 3x 6 0
 1.6.15: x4 x3 x 1 0 x
 1.6.16: x4 2x3 8x 16 0 x
 1.6.17: 4x2 3 0
 1.6.18: x4 5x x 2 36 0 4
 1.6.19: 4x 2 7 0 4 65x 2 16 0 x
 1.6.20: 36t4 29t 4x 2 7 0 4
 1.6.21: x 3 2 0 6 7x3 8 0
 1.6.22: x6 3x x 3 2 0
 1.6.23: 1 x 2 8 x 15 0
 1.6.24: 1 3 x 2 x2
 1.6.25: 2 x x 2 2 3 x x 2 2 0 1
 1.6.26: 6 x x 1 2 5 x x 1 6 0
 1.6.27: 2x 9x 5 6
 1.6.28: 6x 7x 3 0 6
 1.6.29: 3x1 3 2x2 3 5
 1.6.30: 9t 2 3 24t1 3 16 0
 1.6.31: x3 2x 2 3x x
 1.6.32: y 2x 4 15x 3 18x2 y
 1.6.33: y x 4 10x 2 9 y
 1.6.34: y x 4 29x 2 100
 1.6.35: 3x 12 0 7x
 1.6.36: 7x 4 0 y
 1.6.37: x 10 4 0 5
 1.6.38: 5 x 3 0 3x
 1.6.39: 3 2x 5 3 0
 1.6.40: 3 3x 1 5 0
 1.6.41: 26 11x 4 x x
 1.6.42: x 31 9x 5
 1.6.43: x 1 3x 1 x
 1.6.44: x 5 x 5 26
 1.6.45: x x 5 1 x
 1.6.46: x x 20 10 x
 1.6.47: x 5 x 5 10 x
 1.6.48: 2x 1 2x 3 1 x
 1.6.49: x 2 2x 3 1 2x
 1.6.50: 4x 3 6x 17 3 x 2
 1.6.51: x 53 2 8
 1.6.52: x 33 2 8
 1.6.53: x 32 3 8
 1.6.54: x 22 3 9
 1.6.55: x2 53 2 27
 1.6.56: x2 x 223 2 27
 1.6.57: 3x x 11 2 2 x 13 2 0
 1.6.58: 4x2 x 11 3 6x x 14 3 0 3
 1.6.59: y 11x 30 x x
 1.6.60: 2x 15 4x y
 1.6.61: y 7x 36 5x 16 2 y
 1.6.62: y 3x 4 x 4 y
 1.6.63:
 1.6.64:
 1.6.65: 1 x
 1.6.66: 4 x 1 3 x 2 1 1
 1.6.67: 30 x x x
 1.6.68: 4x 1 3 x
 1.6.69: x x 2 4 1 x 2 3 4
 1.6.70: x 1 3 x 1 x 2 0
 1.6.71: 2x 5 11 x
 1.6.72: 3x 2 7
 1.6.73: x x2 x 24
 1.6.74: x 2 6x 3x 18
 1.6.75: x 1 x 2 5
 1.6.76: x 15 x 2 15x
 1.6.77: y 1 x 4 x 1 1 x
 1.6.78: y x 9 x 1 5
 1.6.79: y x 1 2
 1.6.80: y x 2 3 y
 1.6.81: 3.2x 4 1.5x2 2.1 0 y
 1.6.82: 0.1x4 2.4x2 3.6 0 3
 1.6.83: 7.08x6 4.15x 3 9.6 0
 1.6.84: 5.25x6 0.2x3 1.55 0 7
 1.6.85: 1.8x 6x 5.6 0 5.
 1.6.86: 2.4x 12.4x 0.28 0 1
 1.6.87: 4x 2 3 8x1 3 3.6 0
 1.6.88: 8.4x2 3 1.2x1 3 24 0 4
 1.6.89: 4, 7
 1.6.90: 2000, 2, 9
 1.6.91: 5 7 3, 6 7
 1.6.92: 1 8, 4 5 7
 1.6.93: 3, 3, 4 27,
 1.6.94: 27, 7 1
 1.6.95: i, i 2
 1.6.96: 2i, 2i 3
 1.6.97: 1, 1, i, i 4
 1.6.98: 4i, 4i, 6, 6 i
 1.6.99: CHARTERING A BUS A college charters a bus for $1700 to take a group...
 1.6.100: RENTING AN APARTMENT Three students are planning to rent an apartme...
 1.6.101: AIRSPEED An airline runs a commuter flight between Portland, Oregon...
 1.6.102: AVERAGE SPEED A family drove 1080 miles to their vacation lodge. Be...
 1.6.103: MUTUAL FUNDS A deposit of $2500 in a mutual fund reaches a balance ...
 1.6.104: MUTUAL FUNDS A sales representative for a mutual funds company desc...
 1.6.105: NUMBER OF DOCTORS The number of medical doctors (in thousands) in t...
 1.6.106: VOTING POPULATION The total votingage population (in millions) in ...
 1.6.107: SATURATED STEAM The temperature (in degrees Fahrenheit) of saturate...
 1.6.108: AIRLINE PASSENGERS An airline offers daily flights between Chicago ...
 1.6.109: DEMAND The demand equation for a video game is modeled by
 1.6.110: DEMAND The demand equation for a high definition television set is ...
 1.6.111: BASEBALL A baseball diamond has the shape of a square in which the ...
 1.6.112: METEOROLOGY A meteorologist is positioned 100 feet from the point w...
 1.6.113: GEOMETRY You construct a cone with a base radius of 8 inches. The l...
 1.6.114: LABOR Working together, two people can complete a task in 8 hours. ...
 1.6.115: LABOR Working together, two people can complete a task in 12 hours....
 1.6.116: POWER LINE A power station is on one side of a river that is mile w...
 1.6.117: A PERSONS TANGENTIAL SPEED IN A ROTOR
 1.6.118: INDUCTANCE
 1.6.119: An equation can never have more than one extraneous solution.
 1.6.120: When solving an absolute value equation, you will always have to ch...
 1.6.121: The equation has no solution.
 1.6.122: CAPSTONE When solving an equation, list three operations that can i...
 1.6.123: 1, 2, x, 10
 1.6.124: 8, 0, x, 5 x
 1.6.125: 0, 0, 8, y
 1.6.126: 8, 4, 7, y
 1.6.127: Find and when the solution of the equation is (There are many corre...
 1.6.128: WRITING Write a short paragraph listing the steps required to solve...
 1.6.129: Find and when the solution of the equation is (There are many corre...
 1.6.130: WRITING Write a short paragraph listing the steps required to solve...
Solutions for Chapter 1.6: Other Types of Equations
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 1.6: Other Types of Equations
Get Full SolutionsAlgebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.6: Other Types of Equations includes 130 full stepbystep solutions. Since 130 problems in chapter 1.6: Other Types of Equations have been answered, more than 46609 students have viewed full stepbystep solutions from this chapter.

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).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

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 •

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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 ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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!

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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

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