 2.1.1: Fill in the blanks. The simplest mathematical model for relating tw...
 2.1.2: Fill in the blanks. For a line, the ratio of the change in to the c...
 2.1.3: Fill in the blanks. Two lines are ________ if and only if their slo...
 2.1.4: Fill in the blanks. Two lines are ________ if and only if their slo...
 2.1.5: Fill in the blanks. When the axis and axis have different units o...
 2.1.6: Fill in the blanks. The prediction method ________ ________ is the ...
 2.1.7: Fill in the blanks. Every line has an equation that can be written ...
 2.1.8: Fill in the blanks. Match each equation of a line with its form. (a...
 2.1.9: In Exercises 9 and 10, identify the line that has each slope.
 2.1.10: In Exercises 9 and 10, identify the line that has each slope.
 2.1.11: In Exercises 11 and 12, sketch the lines through the point with the...
 2.1.12: In Exercises 11 and 12, sketch the lines through the point with the...
 2.1.13: In Exercises 1316, estimate the slope of the line.
 2.1.14: In Exercises 1316, estimate the slope of the line.
 2.1.15: In Exercises 1316, estimate the slope of the line.
 2.1.16: In Exercises 1316, estimate the slope of the line.
 2.1.17: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.18: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.19: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.20: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.21: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.22: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.23: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.24: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.25: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.26: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.27: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.28: In Exercises 1728, find the slope and intercept (if possible) of t...
 2.1.29: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.30: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.31: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.32: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.33: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.34: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.35: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.36: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.37: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.38: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.39: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.40: In Exercises 2940, plot the points and find the slope of the line p...
 2.1.41: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.42: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.43: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.44: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.45: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.46: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.47: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.48: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.49: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.50: In Exercises 4150, use the point on the line and the slope of the l...
 2.1.51: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.52: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.53: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.54: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.55: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.56: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.57: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.58: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.59: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.60: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.61: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.62: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.63: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.64: In Exercises 51 64, find the slopeintercept form of the equation o...
 2.1.65: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.66: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.67: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.68: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.69: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.70: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.71: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.72: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.73: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.74: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.75: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.76: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.77: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.78: In Exercises 6578, find the slopeintercept form of the equation of...
 2.1.79: In Exercises 79 82, determine whether the lines are parallel, perpe...
 2.1.80: In Exercises 79 82, determine whether the lines are parallel, perpe...
 2.1.81: In Exercises 79 82, determine whether the lines are parallel, perpe...
 2.1.82: In Exercises 79 82, determine whether the lines are parallel, perpe...
 2.1.83: In Exercises 8386, determine whether the lines and passing through ...
 2.1.84: In Exercises 8386, determine whether the lines and passing through ...
 2.1.85: In Exercises 8386, determine whether the lines and passing through ...
 2.1.86: In Exercises 8386, determine whether the lines and passing through ...
 2.1.87: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.88: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.89: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.90: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.91: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.92: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.93: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.94: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.95: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.96: In Exercises 8796, write the slopeintercept forms of the equations...
 2.1.97: In Exercises 97102, use the intercept form to find the equation of ...
 2.1.98: In Exercises 97102, use the intercept form to find the equation of ...
 2.1.99: In Exercises 97102, use the intercept form to find the equation of ...
 2.1.100: In Exercises 97102, use the intercept form to find the equation of ...
 2.1.101: In Exercises 97102, use the intercept form to find the equation of ...
 2.1.102: In Exercises 97102, use the intercept form to find the equation of ...
 2.1.103: GRAPHICAL ANALYSIS In Exercises 103106, identify any relationships ...
 2.1.104: GRAPHICAL ANALYSIS In Exercises 103106, identify any relationships ...
 2.1.105: GRAPHICAL ANALYSIS In Exercises 103106, identify any relationships ...
 2.1.106: GRAPHICAL ANALYSIS In Exercises 103106, identify any relationships ...
 2.1.107: In Exercises 107110, find a relationship between and such that is e...
 2.1.108: In Exercises 107110, find a relationship between and such that is e...
 2.1.109: In Exercises 107110, find a relationship between and such that is e...
 2.1.110: In Exercises 107110, find a relationship between and such that is e...
 2.1.111: SALES The following are the slopes of lines representing annual sal...
 2.1.112: REVENUE The following are the slopes of lines representing daily re...
 2.1.113: AVERAGE SALARY The graph shows the average salaries for senior high...
 2.1.114: SALES The graph shows the sales (in billions of dollars) for Apple ...
 2.1.115: ROAD GRADE You are driving on a road that has a 6% uphill grade (se...
 2.1.116: ROAD GRADE From the top of a mountain road, a surveyor takes severa...
 2.1.117: $2540 $125 decrease per year
 2.1.118: $156 $4.50 increase per year
 2.1.119: DEPRECIATION The value of a molding machine years after it is purch...
 2.1.120: COST The cost of producing computer laptop bags is given by Explain...
 2.1.121: DEPRECIATION A sub shop purchases a used pizza oven for $875. After...
 2.1.122: DEPRECIATION A school district purchases a highvolume printer, cop...
 2.1.123: SALES A discount outlet is offering a 20% discount on all items. Wr...
 2.1.124: HOURLY WAGE A microchip manufacturer pays its assembly line workers...
 2.1.125: MONTHLY SALARY A pharmaceutical salesperson receives a monthly sala...
 2.1.126: BUSINESS COSTS A sales representative of a company using a personal...
 2.1.127: CASH FLOW PER SHARE The cash flow per share for the Timberland Co. ...
 2.1.128: NUMBER OF STORES In 2003 there were 1078 J.C. Penney stores and in ...
 2.1.129: COLLEGE ENROLLMENT The Pennsylvania State University had enrollment...
 2.1.130: COLLEGE ENROLLMENT The University of Florida had enrollments of 46,...
 2.1.131: COST, REVENUE, AND PROFIT A roofing contractor purchases a shingle ...
 2.1.132: RENTAL DEMAND A real estate office handles an apartment complex wit...
 2.1.133: GEOMETRY The length and width of a rectangular garden are 15 meters...
 2.1.134: AVERAGE ANNUAL SALARY The average salaries (in millions of dollars)...
 2.1.135: DATA ANALYSIS: NUMBER OF DOCTORS The numbers of doctors of osteopat...
 2.1.136: DATA ANALYSIS: AVERAGE SCORES An instructor gives regular 20point ...
 2.1.137: A line with a slope of is steeper than a line with a slope of
 2.1.138: The line through and and the line through and are parallel.
 2.1.139: Explain how you could show that the points and are the vertices of ...
 2.1.140: Explain why the slope of a vertical line is said to be undefined.
 2.1.141: With the information shown in the graphs, is it possible to determi...
 2.1.142: The slopes of two lines are and Which is steeper? Explain.
 2.1.143: Use a graphing utility to compare the slopes of the lines where 1, ...
 2.1.144: Find and in terms of and respectively (see figure). Then use the Py...
 2.1.145: THINK ABOUT IT Is it possible for two lines with positive slopes to...
 2.1.146: CAPSTONE Match the description of the situation with its graph. Als...
Solutions for Chapter 2.1: LINEAR EQUATIONS IN TWO VARIABLES
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 2.1: LINEAR EQUATIONS IN TWO VARIABLES
Get Full SolutionsChapter 2.1: LINEAR EQUATIONS IN TWO VARIABLES includes 146 full stepbystep solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 8. College Algebra was written by and is associated to the ISBN: 9781439048696. Since 146 problems in chapter 2.1: LINEAR EQUATIONS IN TWO VARIABLES have been answered, more than 33191 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

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 .

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).