 6.7.1: The first step in solving 1 x + 1 y = 1 z for z is to multiply both...
 6.7.2: The cost function for a business, C(x), is the sum of the _________...
 6.7.3: The average cost function for a company, C(x), is its cost function...
 6.7.4: The formula t = d r states that time traveled is ______________ div...
 6.7.5: In work problems, the number ______________ represents the whole jo...
 6.7.6: If you can complete a job in 19 hours, the fractional part of the j...
 6.7.7: F = Gm1m2 d2 for m1 (physics)
 6.7.8: F = Gm1m2 d2 for m2 (physics)
 6.7.9: z = x  x s for x (statistics)
 6.7.10: z = x  x s for s (statistics)
 6.7.11: I = E R + r for R (electronics)
 6.7.12: I = E R + r for r (electronics)
 6.7.13: f = f1 f2 f1 + f2 for f1 (optics)
 6.7.14: f = f1 f2 f1 + f2 for f2 (optics)
 6.7.15: How many wheelchairs must be produced each month for the company to...
 6.7.16: How many wheelchairs must be produced each month for the company to...
 6.7.17: What is the equation of the horizontal asymptote shown by the dashe...
 6.7.18: Describe the end behavior of the graph at the far right. Is there a...
 6.7.19: A company is planning to manufacture mountain bikes. Fixed monthly ...
 6.7.20: A company is planning to manufacture small canoes. Fixed monthly co...
 6.7.21: If the total time for driving and running is 3 hours, what is your ...
 6.7.22: If the total time for driving and running is 5 hours, what is your ...
 6.7.23: Describe the behavior of the graph as x approaches 0. What does thi...
 6.7.24: The graph is falling from left to right. What does this show?
 6.7.25: A car can travel 300 miles in the same amount of time it takes a bu...
 6.7.26: A passenger train can travel 240 miles in the same amount of time i...
 6.7.27: You ride your bike to campus a distance of 5 miles and return home ...
 6.7.28: An engine pulls a train 140 miles. Then a second engine, whose aver...
 6.7.29: In still water, a boat averages 7 miles per hour. It takes the same...
 6.7.30: In still water, a boat averages 8 miles per hour. It takes the same...
 6.7.31: The rate of the jet stream is 100 miles per hour. Traveling with th...
 6.7.32: The wind is blowing at an average rate of 10 miles per hour. Riding...
 6.7.33: A moving sidewalk at an airport glides at a rate of 1.8 feet per se...
 6.7.34: A moving sidewalk at an airport glides at a rate of 1.8 feet per se...
 6.7.35: Two runners, one averaging 8 miles per hour and the other 6 miles p...
 6.7.36: Two sailboats, one averaging 20 miles per hour and the other 18 mil...
 6.7.37: You promised your parents that you would wash the family car. You h...
 6.7.38: You must leave for campus in half an hour, or you will be late for ...
 6.7.39: A pool can be filled by one pipe in 6 hours and by a second pipe in...
 6.7.40: A pond can be filled by one pipe in 8 hours and by a second pipe in...
 6.7.41: Working with your cousin, you can refinish a table in 3 hours. Work...
 6.7.42: Working with your cousin, you can split a cord of firewood in 5 hou...
 6.7.43: An earthquake strikes and an isolated area is without food or water...
 6.7.44: A hurricane strikes and a rural area is without food or water. Thre...
 6.7.45: An office has an old copying machine and a new one. Working togethe...
 6.7.46: A demolition company wants to build a brick wall to hide from publi...
 6.7.47: A faucet can fill a sink in 5 minutes. It takes twice as long for t...
 6.7.48: A pool can be filled by a pipe in 3 hours. It takes 3 times as long...
 6.7.49: What number multiplied by the numerator and added to the denominato...
 6.7.50: What number multiplied by the numerator and subtracted from the den...
 6.7.51: The sum of 2 times a number and twice its reciprocal is 20 3 . Find...
 6.7.52: If 2 times the reciprocal of a number is subtracted from 3 times th...
 6.7.53: You have 35 hits in 140 times at bat. Your batting average is 35 14...
 6.7.54: You have 30 hits in 120 times at bat. Your batting average is 30 12...
 6.7.55: If one pipe can fill a pool in a hours and a second pipe can fill t...
 6.7.56: If one pipe can fill a pool in a hours and a second pipe can empty ...
 6.7.57: Without showing the details, explain how to solve the formula 1 R =...
 6.7.58: Explain how to find the average cost function for a business.
 6.7.59: How does the average cost function illustrate a problem for small b...
 6.7.60: What is the relationship among time traveled, distance traveled, an...
 6.7.61: If you know how many hours it takes for you to do a job, explain ho...
 6.7.62: If you can do a job in 6 hours and your friend can do the same job ...
 6.7.63: When two people work together to complete a job, describe one facto...
 6.7.64: For Exercises 1920, use a graphing utility to graph the average cos...
 6.7.65: For Exercises 4546, use a graphing utility to graph the function re...
 6.7.66: A boat can travel 10 miles per hour in still water. The boat travel...
 6.7.67: I decided to organize the critical information from the advertiseme...
 6.7.68: The ad stated that bikes with coverings reduced time on the 75mile...
 6.7.69: The equation in x that modeled the conditions had a positive and a ...
 6.7.70: My professor verified that 15 is the correct value for x in the equ...
 6.7.71: As production level increases, the average cost for a company to pr...
 6.7.72: To solve qf + pf = pq for p, subtract qf from both sides and then d...
 6.7.73: If you plan a theater trip that costs $300 to rent a limousine and ...
 6.7.74: If you can clean the house in 3 hours and your sloppy friend can co...
 6.7.75: Solve 1 s = f + 1  f p for f
 6.7.76: A new schedule for a train requires it to travel 351 miles in 1 4 h...
 6.7.77: It takes Mr. Todd 4 hours longer to prepare an order of pies than i...
 6.7.78: Factor: x2 + 4x + 4  9y2 . (Section 5.6, Example 4)
 6.7.79: Solve using matrices: e 2x + 5y = 5 x + 2y = 1. (Section 3.4, Exa...
 6.7.80: Solve the system: x + y + z = 4 2x + 5y = 1 x  y  2x = 0. (Sectio...
 6.7.81: a. If y = kx2 , find the value of k using x = 2 and y = 64. b. Subs...
 6.7.82: a. If y = k x , find the value of k using x = 8 and y = 12. b. Subs...
 6.7.83: If S = kA P , find the value of k using A = 60,000, P = 40, and S =...
Solutions for Chapter 6.7: Formulas and Applications of Rational Equations
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 6.7: Formulas and Applications of Rational Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.7: Formulas and Applications of Rational Equations includes 83 full stepbystep solutions. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Since 83 problems in chapter 6.7: Formulas and Applications of Rational Equations have been answered, more than 15983 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

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

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

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.
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