 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 34119 students have viewed full stepbystep solutions from this chapter. Chapter 8.1: Systems of Linear Equations: Substitution and Elimination includes 84 full stepbystep solutions.

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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 .

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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 .

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).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

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

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.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.