 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 36854 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.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

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.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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 row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

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.

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

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

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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