 2.1.1: True or False The lines and are parallel.
 2.1.2: Solve: 2x  6 = 6  4x
 2.1.3: Find the point of intersection of the lines y = 60x  900 y = 15x ...
 2.1.4: Solve for x if 2x + 3z = 6
 2.1.5: True or False A system of two linear equations containing two varia...
 2.1.6: A system of equations that has no solution is called _____.
 2.1.7: In 714, decide whether the values of the variables listed are solut...
 2.1.8: In 714, decide whether the values of the variables listed are solut...
 2.1.9: In 714, decide whether the values of the variables listed are solut...
 2.1.10: In 714, decide whether the values of the variables listed are solut...
 2.1.11: In 714, decide whether the values of the variables listed are solut...
 2.1.12: In 714, decide whether the values of the variables listed are solut...
 2.1.13: In 714, decide whether the values of the variables listed are solut...
 2.1.14: In 714, decide whether the values of the variables listed are solut...
 2.1.15: In 1542, solve each system of equations. If the system has no solut...
 2.1.16: In 1542, solve each system of equations. If the system has no solut...
 2.1.17: In 1542, solve each system of equations. If the system has no solut...
 2.1.18: In 1542, solve each system of equations. If the system has no solut...
 2.1.19: In 1542, solve each system of equations. If the system has no solut...
 2.1.20: In 1542, solve each system of equations. If the system has no solut...
 2.1.21: In 1542, solve each system of equations. If the system has no solut...
 2.1.22: In 1542, solve each system of equations. If the system has no solut...
 2.1.23: In 1542, solve each system of equations. If the system has no solut...
 2.1.24: In 1542, solve each system of equations. If the system has no solut...
 2.1.25: In 1542, solve each system of equations. If the system has no solut...
 2.1.26: In 1542, solve each system of equations. If the system has no solut...
 2.1.27: In 1542, solve each system of equations. If the system has no solut...
 2.1.28: In 1542, solve each system of equations. If the system has no solut...
 2.1.29: In 1542, solve each system of equations. If the system has no solut...
 2.1.30: In 1542, solve each system of equations. If the system has no solut...
 2.1.31: In 1542, solve each system of equations. If the system has no solut...
 2.1.32: In 1542, solve each system of equations. If the system has no solut...
 2.1.33: In 1542, solve each system of equations. If the system has no solut...
 2.1.34: In 1542, solve each system of equations. If the system has no solut...
 2.1.35: In 1542, solve each system of equations. If the system has no solut...
 2.1.36: In 1542, solve each system of equations. If the system has no solut...
 2.1.37: In 1542, solve each system of equations. If the system has no solut...
 2.1.38: In 1542, solve each system of equations. If the system has no solut...
 2.1.39: In 1542, solve each system of equations. If the system has no solut...
 2.1.40: In 1542, solve each system of equations. If the system has no solut...
 2.1.41: In 1542, solve each system of equations. If the system has no solut...
 2.1.42: In 1542, solve each system of equations. If the system has no solut...
 2.1.43: Dimensions of a Floor The perimeter of a rectangular floor is 90 fe...
 2.1.44: Dimensions of a Field The length of fence required to enclose a rec...
 2.1.45: Agriculture According to the Illinois Farm Business Management Asso...
 2.1.46: Movie Theater Tickets The Coral movie theater charges $9.00 for adu...
 2.1.47: Mixing Nuts A store sells cashews for $5.00 per pound and peanuts f...
 2.1.48: Financial Planning A recently retired couple needs $6000 per year t...
 2.1.49: Cost of Food in Japan In Osaka, Japan, the cost of three 1liter ca...
 2.1.50: Cost of Fast Food Four large cheeseburgers and two chocolate shakes...
 2.1.51: Computing a Refund The grocery store we use does not mark prices on...
 2.1.52: Blending Coffees A coffee manufacturer wants to market a new blend ...
 2.1.53: Pharmacy A doctors prescription calls for a daily intake of liquid ...
 2.1.54: Pharmacy A doctors prescription calls for the creation of pills tha...
 2.1.55: Diet Preparation A 600 to 700pound yearling horse needs 33.0 gram...
 2.1.56: Restaurant Management A restaurant manager wants to purchase 200 se...
 2.1.57: Theater Revenues A Broadway theater has 500 seats, divided into orc...
 2.1.58: Theater Revenues The Star movie theater charges $8.00 for adults, $...
 2.1.59: Prices of Fast Food One group of customers bought 8 deluxe hamburge...
 2.1.60: Prices of Fast Food Use the information given in 59. Suppose that a...
 2.1.61: Supply and Demand Suppose that the quantity supplied S and quantity...
 2.1.62: Supply and Demand Suppose that the quantity supplied S and quantity...
 2.1.63: ISLM Model in Economics In economics, the IS curve is a linear equ...
 2.1.64: ISLM Model in Economics In economics, the IS curve is a linear equ...
 2.1.65: Financial Planning Nathan wants to invest long term the $70,000 pro...
 2.1.66: Financial Planning A married couple wants to invest $60,000 long te...
 2.1.67: Make up three systems of two linear equations containing two variab...
 2.1.68: Write a brief paragraph outlining your strategy for solving a syste...
 2.1.69: Do you prefer the method of substitution or the method of eliminati...
 2.1.70: Look at the table obtained in Example 14. What advice would you giv...
Solutions for Chapter 2.1: Systems of Linear Equations: Substitution; Elimination
Full solutions for Finite Mathematics, Binder Ready Version: An Applied Approach  11th Edition
ISBN: 9780470876398
Solutions for Chapter 2.1: Systems of Linear Equations: Substitution; Elimination
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.1: Systems of Linear Equations: Substitution; Elimination includes 70 full stepbystep solutions. This textbook survival guide was created for the textbook: Finite Mathematics, Binder Ready Version: An Applied Approach, edition: 11. Finite Mathematics, Binder Ready Version: An Applied Approach was written by and is associated to the ISBN: 9780470876398. Since 70 problems in chapter 2.1: Systems of Linear Equations: Substitution; Elimination have been answered, more than 16741 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

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

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

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.

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.