 8.1.1: Solve the equation: 3x + 4 = 8  x.
 8.1.2: (a) Graph the line: 3x + 4y = 12 (b) What is the slope of a line pa...
 8.1.3: If a system of equations has no solution, it is said to be .
 8.1.4: If a system of equations has one solution, the system is and the eq...
 8.1.5: If the solution to a system of two linear equations containing two ...
 8.1.6: If the lines that make up a system of two linear equations are coin...
 8.1.7: e 2x  y = 5 5x + 2y = 8 x = 2, y = 1; 12, 12
 8.1.8: b 3x + 2y = 2 x  7y = 30 x = 2, y = 4; 12, 42
 8.1.9: c 3x  4y = 4 1 2 x  3y =  1 2 x = 2, y = 12; a2, 12b
 8.1.10: d 2x + 1 2 y = 3x  4y = 0  19 2 x =  12, y = 2; a 12, 2b
 8.1.11: L x  y = 3 1 2 x + y = 3 x = 4, y = 1; 14, 12
 8.1.12: e x  y = 3 3x + y = 1 x = 2, y = 5; 12, 52
 8.1.13: c 3x + 3y + 2z = 4 x  y  z = 0 2y  3z = 8 x = 1, y = 1, z = 21...
 8.1.14: c 4x  z = 7 8x + 5y  z = 0 x  y + 5z = 6 x = 2, y = 3, z = 1 1...
 8.1.15: c 3x + 3y + 2z = 4 x  3y + z = 10 5x  2y  3z = 8 x = 2, y = 2, ...
 8.1.16: c 4x  5z = 6 5y  z = 17 x  6y + 5z = 24 x = 4, y = 3, z = 2; ...
 8.1.17: b x + y = 8 x  y = 4
 8.1.18: b x + 2y = 7 x + y = 3
 8.1.19: 5x  y = 21 2x + 3y = 12
 8.1.20: b x + 3y = 5 2x  3y = 8
 8.1.21: b 3x = 24 x + 2y = 0
 8.1.22: b 4x + 5y = 3 2y = 8
 8.1.23: b 3x  6y = 2 5x + 4y = 1
 8.1.24: c 2x + 4y = 2 3 3x  5y = 10
 8.1.25: b 2x + y = 1 4x + 2y = 3
 8.1.26: b x  y = 5 3x + 3y = 2
 8.1.27: b 2x  y = 0 4x + 2y = 12
 8.1.28: c 3x + 3y = 1 4x + y = 8 3
 8.1.29: b x + 2y = 4 2x + 4y = 8
 8.1.30: b 3x  y = 7 9x  3y = 21
 8.1.31: b 2x  3y = 1 10x + y = 11
 8.1.32: b 3x  2y = 0 5x + 10y = 4
 8.1.33: c 2x + 3y = 6 x  y = 1 2
 8.1.34: c 1 2 x + y = 2 x  2y = 8
 8.1.35: 1 2 x + 1 3 y = 3 1 4 x  2 3 y = 1
 8.1.36: d 1 3 x  3 2 y = 3 4 x + 1 3 y = 5 11
 8.1.37: b 3x  5y = 3 15x + 5y = 21
 8.1.38: c 2x  y = 1 x + 1 2 y = 3 2
 8.1.39: d 1 x + 1 y = 8 3 x  5 y = 0
 8.1.40: d 4 x  3 y = 0 6 x + 3 2y = 2
 8.1.41: c x  y = 6 2x  3z = 16 2y + z = 4
 8.1.42: c 2x + y = 4 2y + 4z = 0 3x  2z = 11
 8.1.43: c x  2y + 3z = 7 2x + y + z = 4 3x + 2y  2z = 10
 8.1.44: c 2x + y  3z = 0 2x + 2y + z = 7 3x  4y  3z = 7
 8.1.45: c x  y  z = 1 2x + 3y + z = 2 3x + 2y = 0
 8.1.46: c 2x  3y  z = 0 x + 2y + z = 5 3x  4y  z = 1
 8.1.47: c x  y  z = 1 x + 2y  3z = 4 3x  2y  7z = 0
 8.1.48: c 2x  3y  z = 0 3x + 2y + 2z = 2 x + 5y + 3z = 2
 8.1.49: c 2x  2y + 3z = 6 4x  3y + 2z = 0 2x + 3y  7z = 1
 8.1.50: c 3x  2y + 2z = 6 7x  3y + 2z = 1 2x  3y + 4z = 0
 8.1.51: c x + y  z = 6 3x  2y + z = 5 x + 3y  2z = 14
 8.1.52: c x  y + z = 4 2x  3y + 4z = 15 5x + y  2z = 12
 8.1.53: c x + 2y  z = 3 2x  4y + z = 7 2x + 2y  3z = 4
 8.1.54: c x + 4y  3z = 8 3x  y + 3z = 12 x + y + 6z = 1
 8.1.55: The perimeter of a rectangular floor is 90 feet. Find the dimension...
 8.1.56: The length of fence required to enclose a rectangular field is 3000...
 8.1.57: Orbital Launches In 2005 there was a total of 55 commercial and non...
 8.1.58: Movie Theater Tickets A movie theater charges $9.00 for adults and ...
 8.1.59: Mixing Nuts A store sells cashews for $5.00 per pound and peanuts f...
 8.1.60: Financial Planning A recently retired couple needs $12,000 per year...
 8.1.61: Computing Wind Speed With a tail wind, a small Piper aircraft can f...
 8.1.62: Computing Wind Speed The average airspeed of a singleengine aircraf...
 8.1.63: Restaurant Management A restaurant manager wants to purchase 200 se...
 8.1.64: Cost of Fast Food One group of people purchased 10 hot dogs and 5 s...
 8.1.65: Computing a Refund The grocery store we use does not mark prices on...
 8.1.66: Finding the Current of a Stream Pamela requires 3 hours to swim 15 ...
 8.1.67: Pharmacy A doctors prescription calls for a daily intake containing...
 8.1.68: Pharmacy A doctors prescription calls for the creation of pills tha...
 8.1.69: Curve Fitting Find real numbers a, b, and c so that the graph of th...
 8.1.70: Curve Fitting Find real numbers a, b, and c so that the graph of th...
 8.1.71: ISLM Model in Economics In economics, the IS curve is a linear equa...
 8.1.72: ISLM Model in Economics In economics, the IS curve is a linear equa...
 8.1.73: Electricity: Kirchhoffs Rules An application of Kirchhoffs Rules to...
 8.1.74: Electricity: Kirchhoffs Rules An application of Kirchhoffs Rules to...
 8.1.75: Theater Revenues A Broadway theater has 500 seats, divided into orc...
 8.1.76: Theater Revenues A movie theater charges $8.00 for adults, $4.50 fo...
 8.1.77: Nutrition A dietitian wishes a patient to have a meal that has 66 g...
 8.1.78: Investments Kelly has $20,000 to invest. As her financial planner, ...
 8.1.79: Prices of Fast Food One group of customers bought 8 deluxe hamburge...
 8.1.80: Prices of Fast Food Use the information given in 79. Suppose that a...
 8.1.81: Painting a House Three painters, Beth, Bill, and Edie, working toge...
 8.1.82: Make up a system of three linear equations containing three variabl...
 8.1.83: Write a brief paragraph outlining your strategy for solving a syste...
 8.1.84: Do you prefer the method of substitution or the method of eliminati...
Solutions for Chapter 8.1: Systems of Linear Equations: Substitution and Elimination
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 8.1: Systems of Linear Equations: Substitution and Elimination
Get Full SolutionsCollege Algebra was written by and is associated to the ISBN: 9780321716811. This textbook survival guide was created for the textbook: College Algebra, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 84 problems in chapter 8.1: Systems of Linear Equations: Substitution and Elimination have been answered, more than 9303 students have viewed full stepbystep solutions from this chapter. Chapter 8.1: Systems of Linear Equations: Substitution and Elimination includes 84 full stepbystep solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

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].

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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

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.

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.

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.

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

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).
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