 1.3.1: The process of translating phrases or sentences into algebraic expr...
 1.3.2: A good approach to mathematical modeling is a twostage approach, u...
 1.3.3: Area of a circle: ________
 1.3.4: Perimeter of a rectangle: ________
 1.3.5: Volume of a cube: ________
 1.3.6: Volume of a circular cylinder: ________
 1.3.7: Balance if dollars is invested at % compounded monthly for years: _...
 1.3.8: Simple interest if dollars is invested at % for years: ________
 1.3.9: x
 1.3.10:
 1.3.11: u 5
 1.3.12: 2 3
 1.3.13: 4 5
 1.3.14: z 10 7
 1.3.15: 3 b 21
 1.3.16: 12x x 5
 1.3.17: 4 p 1 p
 1.3.18: q 4 3 q 2q
 1.3.19: The sum of two consecutive natural numbers
 1.3.20: The product of two consecutive natural numbers
 1.3.21: The product of two consecutive odd integers, the first of which is
 1.3.22: The sum of the squares of two consecutive even integers, the first ...
 1.3.23: The distance traveled in hours by a car traveling at 55 miles per hour
 1.3.24: The travel time for a plane traveling at a rate of kilometers per h...
 1.3.25: The amount of acid in liters of a 20% acid solution
 1.3.26: The sale price of an item that is discounted 33% of its list price
 1.3.27: The perimeter of a rectangle with a width and a length that is twic...
 1.3.28: The area of a triangle with base 16 inches and height inches
 1.3.29: The total cost of producing units for which the fixed costs are $25...
 1.3.30: The total revenue obtained by selling units at $12.99 per unit
 1.3.31: Thirty percent of the list price
 1.3.32: The amount of water in quarts of a liquid that is 28% water
 1.3.33: The percent of 672 that is represented by the number
 1.3.34: The percent change in sales from one month to the next if the month...
 1.3.35: In Exercises 35 and 36, write an expression for the area of the reg...
 1.3.36: In Exercises 35 and 36, write an expression for the area of the reg...
 1.3.37: The sum of two consecutive natural numbers is 525. Find the numbers.
 1.3.38: The sum of three consecutive natural numbers is 804. Find the numbers.
 1.3.39: One positive number is 5 times another number. The difference betwe...
 1.3.40: One positive number is of another number. The difference between th...
 1.3.41: Find two consecutive integers whose product is 5 less than the squa...
 1.3.42: Find two consecutive natural numbers such that the difference of th...
 1.3.43: What is 30% of 45?
 1.3.44: What is 175% of 360?
 1.3.45: 432 is what percent of 1600?
 1.3.46: 459 is what percent of 340?
 1.3.47: 12 is of what number?
 1.3.48: 70 is 40% of what number?
 1.3.49: FINANCE A salespersons weekly paycheck is 15% less than a second sa...
 1.3.50: DISCOUNT The price of a swimming pool has been discounted 16.5%. Th...
 1.3.51: FINANCE A family has annual loan payments equaling 32% of their ann...
 1.3.52: FINANCE A family has a monthly mortgage payment of $500, which is 1...
 1.3.53: Gallon of regular unleaded $1.51 $2.80 gasoline
 1.3.54: Monthly cable rate $30.37 $42.72
 1.3.55: Pound of 100% ground beef $1.63 $2.23
 1.3.56: Monthly bill for cellular $45.27 $49.79 phone service
 1.3.57: DIMENSIONS OF A ROOM A room is 1.5 times as long as it is wide, and...
 1.3.58: DIMENSIONS OF A PICTURE FRAME A picture frame has a total perimeter...
 1.3.59: COURSE GRADE To get an A in a course, you must have an average of a...
 1.3.60: COURSE GRADE You are taking a course that has four tests. The first...
 1.3.61: TRAVEL TIME You are driving on a Canadian freeway to a town that is...
 1.3.62: TRAVEL TIME Students are traveling in two cars to a football game 1...
 1.3.63: AVERAGE SPEED A truck driver traveled at an average speed of 55 mil...
 1.3.64: WIND SPEED An executive flew in the corporate jet to a meeting in a...
 1.3.65: PHYSICS Light travels at the speed of approximately meters per seco...
 1.3.66: RADIO WAVES Radio waves travel at the same speed as light, approxim...
 1.3.67: HEIGHT OF A BUILDING To obtain the height of the Chrysler Building ...
 1.3.68: HEIGHT OF A TREE To obtain the height of a tree (see figure), you m...
 1.3.69: FLAGPOLE HEIGHT A person who is 6 feet tall walks away from a flagp...
 1.3.70: SHADOW LENGTH A person who is 6 feet tall walks away from a 50foot...
 1.3.71: INVESTMENT You plan to invest $12,000 in two funds paying and 5% si...
 1.3.72: INVESTMENT You plan to invest $25,000 in two funds paying 3% and si...
 1.3.73: INVENTORY A nursery has $40,000 of inventory in dogwood trees and r...
 1.3.74: INVENTORY An automobile dealer has $600,000 of inventory in minivan...
 1.3.75: MIXTURE PROBLEM Using the values in the table, determine the amount...
 1.3.76: MIXTURE PROBLEM A 100% concentrate is to be mixed with a mixture ha...
 1.3.77: MIXTURE PROBLEM A forester mixes gasoline and oil to make 2 gallons...
 1.3.78: MIXTURE PROBLEM A grocer mixes peanuts that cost $1.49 per pound an...
 1.3.79: COMPANY COSTS An outdoor furniture manufacturer has fixed costs of ...
 1.3.80: COMPANY COSTS A plumbing supply company has fixed costs of $10,000 ...
 1.3.81: AREA OF A TRIANGLE
 1.3.82: AREA OF A TRAPEZOID
 1.3.83: MARKUP
 1.3.84: INVESTMENT AT SIMPLE INTEREST
 1.3.85: VOLUME OF AN OBLATE SPHEROID
 1.3.86: VOLUME OF A SPHERICAL SEGMENT
 1.3.87: FREEFALLING BODY
 1.3.88: LENSMAKERS EQUATION
 1.3.89: CAPACITANCE IN SERIES CIRCUITS Solve for
 1.3.90: ARITHMETIC PROGRESSION
 1.3.91: ARITHMETIC PROGRESSION
 1.3.92: GEOMETRIC PROGRESSION
 1.3.93: Two children weighing 50 pounds and 75 pounds are playing on a sees...
 1.3.94: A person weighing 200 pounds is attempting to move a 550pound rock...
 1.3.95: VOLUME OF A BILLIARD BALL A billiard ball has a volume of 5.96 cubi...
 1.3.96: LENGTH OF A TANK The diameter of a cylindrical propane gas tank is ...
 1.3.97: TEMPERATURE The average daily temperature in San Diego, California ...
 1.3.98: TEMPERATURE The average daily temperature in Duluth, Minnesota is 3...
 1.3.99: TEMPERATURE The highest temperature ever recorded in Phoenix, Arizo...
 1.3.100: TEMPERATURE The lowest temperature ever recorded in Louisville, Ken...
 1.3.101: 8 less than cubed divided by the difference of squared and 9 can be...
 1.3.102: The volume of a cube with a side of length 9.5 inches is greater th...
 1.3.103: Consider the linear equation (a) What is the sign of the solution i...
 1.3.104: CAPSTONE
 1.3.105: Write a linear equation that has the solution (There are many corre...
Solutions for Chapter 1.3: Modeling with Linear Equations
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 1.3: Modeling with Linear Equations
Get Full SolutionsSince 105 problems in chapter 1.3: Modeling with Linear Equations have been answered, more than 47506 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. This 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. Chapter 1.3: Modeling with Linear Equations includes 105 full stepbystep solutions.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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!).

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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).

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.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.