 2.3.1: The slope of a vertical line is ; the slope of a horizontal line is .
 2.3.2: For the line , the xintercept is and the yintercept is .
 2.3.3: A horizontal line is given by an equation of the form, where b is t...
 2.3.4: True or False Vertical lines have an undefined slope.
 2.3.5: True or False The slope of the line is 3.
 2.3.6: True or False The point is on the line . 11, 22 2x + y = 4 2y = 3x ...
 2.3.7: Two nonvertical lines have slopes and respectively. The lines are p...
 2.3.8: The lines and are parallel if.
 2.3.9: The lines and are perpendicular if.
 2.3.10: True or False Perpendicular lines have slopes that are reciprocals ...
 2.3.11: In 1114, (a) find the slope of the line and (b) interpret the slope.
 2.3.12: In 1114, (a) find the slope of the line and (b) interpret the slope.
 2.3.13: In 1114, (a) find the slope of the line and (b) interpret the slope.
 2.3.14: In 1114, (a) find the slope of the line and (b) interpret the slope.
 2.3.15: In 1522, plot each pair of points and determine the slope of the li...
 2.3.16: In 1522, plot each pair of points and determine the slope of the li...
 2.3.17: In 1522, plot each pair of points and determine the slope of the li...
 2.3.18: In 1522, plot each pair of points and determine the slope of the li...
 2.3.19: In 1522, plot each pair of points and determine the slope of the li...
 2.3.20: In 1522, plot each pair of points and determine the slope of the li...
 2.3.21: In 1522, plot each pair of points and determine the slope of the li...
 2.3.22: In 1522, plot each pair of points and determine the slope of the li...
 2.3.23: In 2330, graph the line containing the point P and having slope m. ...
 2.3.24: In 2330, graph the line containing the point P and having slope m. ...
 2.3.25: In 2330, graph the line containing the point P and having slope m. ...
 2.3.26: In 2330, graph the line containing the point P and having slope m. ...
 2.3.27: In 2330, graph the line containing the point P and having slope m. ...
 2.3.28: In 2330, graph the line containing the point P and having slope m. ...
 2.3.29: In 2330, graph the line containing the point P and having slope m. ...
 2.3.30: In 2330, graph the line containing the point P and having slope m. ...
 2.3.31: In 3136, the slope and a point on a line are given. Use this inform...
 2.3.32: In 3136, the slope and a point on a line are given. Use this inform...
 2.3.33: In 3136, the slope and a point on a line are given. Use this inform...
 2.3.34: In 3136, the slope and a point on a line are given. Use this inform...
 2.3.35: In 3136, the slope and a point on a line are given. Use this inform...
 2.3.36: In 3136, the slope and a point on a line are given. Use this inform...
 2.3.37: In 3744, find an equation of the line L.
 2.3.38: In 3744, find an equation of the line L.
 2.3.39: In 3744, find an equation of the line L.
 2.3.40: In 3744, find an equation of the line L.
 2.3.41: In 3744, find an equation of the line L.
 2.3.42: In 3744, find an equation of the line L.
 2.3.43: In 3744, find an equation of the line L.
 2.3.44: In 3744, find an equation of the line L.
 2.3.45: In 4570, find an equation for the line with the given properties. E...
 2.3.46: In 4570, find an equation for the line with the given properties. E...
 2.3.47: In 4570, find an equation for the line with the given properties. E...
 2.3.48: In 4570, find an equation for the line with the given properties. E...
 2.3.49: In 4570, find an equation for the line with the given properties. E...
 2.3.50: In 4570, find an equation for the line with the given properties. E...
 2.3.51: In 4570, find an equation for the line with the given properties. E...
 2.3.52: In 4570, find an equation for the line with the given properties. E...
 2.3.53: In 4570, find an equation for the line with the given properties. E...
 2.3.54: In 4570, find an equation for the line with the given properties. E...
 2.3.55: In 4570, find an equation for the line with the given properties. E...
 2.3.56: In 4570, find an equation for the line with the given properties. E...
 2.3.57: In 4570, find an equation for the line with the given properties. E...
 2.3.58: In 4570, find an equation for the line with the given properties. E...
 2.3.59: In 4570, find an equation for the line with the given properties. E...
 2.3.60: In 4570, find an equation for the line with the given properties. E...
 2.3.61: In 4570, find an equation for the line with the given properties. E...
 2.3.62: In 4570, find an equation for the line with the given properties. E...
 2.3.63: In 4570, find an equation for the line with the given properties. E...
 2.3.64: In 4570, find an equation for the line with the given properties. E...
 2.3.65: In 4570, find an equation for the line with the given properties. E...
 2.3.66: In 4570, find an equation for the line with the given properties. E...
 2.3.67: In 4570, find an equation for the line with the given properties. E...
 2.3.68: In 4570, find an equation for the line with the given properties. E...
 2.3.69: In 4570, find an equation for the line with the given properties. E...
 2.3.70: In 4570, find an equation for the line with the given properties. E...
 2.3.71: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.72: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.73: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.74: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.75: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.76: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.77: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.78: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.79: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.80: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.81: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.82: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.83: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.84: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.85: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.86: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.87: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.88: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.89: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.90: In 7190, find the slope and yintercept of each line. Graph the lin...
 2.3.91: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.92: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.93: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.94: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.95: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.96: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.97: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.98: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.99: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.100: In 91100, (a) find the intercepts of the graph of each equation and...
 2.3.101: Find an equation of the xaxis
 2.3.102: Find an equation of the yaxis.
 2.3.103: In 103106, the equations of two lines are given. Determine if the l...
 2.3.104: In 103106, the equations of two lines are given. Determine if the l...
 2.3.105: In 103106, the equations of two lines are given. Determine if the l...
 2.3.106: In 103106, the equations of two lines are given. Determine if the l...
 2.3.107: In 107110, write an equation of each line. Express your answer usin...
 2.3.108: In 107110, write an equation of each line. Express your answer usin...
 2.3.109: In 107110, write an equation of each line. Express your answer usin...
 2.3.110: In 107110, write an equation of each line. Express your answer usin...
 2.3.111: Geometry Use slopes to show that the triangle whose vertices are , ...
 2.3.112: Geometry Use slopes to show that the quadrilateral whose vertices a...
 2.3.113: Geometry Use slopes to show that the quadrilateral whose vertices a...
 2.3.114: Geometry Use slopes and the distance formula to show that the quadr...
 2.3.115: Truck Rentals A truck rental company rents a moving truck for one d...
 2.3.116: Cost Equation The fixed costs of operating a business are the costs...
 2.3.117: Cost of Driving a Car The annual fixed costs for owning a small sed...
 2.3.118: Wages of a Car Salesperson Dan receives $375 per week for selling n...
 2.3.119: Electricity Rates in Illinois Commonwealth Edison Company supplies ...
 2.3.120: Electricity Rates in Florida Florida Power & Light Company supplies...
 2.3.121: Measuring Temperature The relationship between Celsius (C) and Fahr...
 2.3.122: Measuring Temperature The Kelvin (K) scale for measuring temperatur...
 2.3.123: Access Ramp A wooden access ramp is being built to reach a platform...
 2.3.124: Cigarette Use A report in the Child Trends DataBase indicated that,...
 2.3.125: Product Promotion A cereal company finds that the number of people ...
 2.3.126: Show that the line containing the points and , , is perpendicular t...
 2.3.127: The equation defines a family of lines, one line for each value of ...
 2.3.128: Prove that if two nonvertical lines have slopes whose product is th...
 2.3.129: Which of the following equations might have the graph shown? (More ...
 2.3.130: Which of the following equations might have the graph shown? (More ...
 2.3.131: The figure shows the graph of two parallel lines.Which of the follo...
 2.3.132: The figure shows the graph of two perpendicular lines. Which of the...
 2.3.133: m is for Slope The accepted symbol used to denote the slope of a li...
 2.3.134: Grade of a Road The term grade is used to describe the inclination ...
 2.3.135: Carpentry Carpenters use the term pitch to describe the steepness o...
 2.3.136: Can the equation of every line be written in slopeintercept form? Why?
 2.3.137: Does every line have exactly one xintercept and one yintercept? A...
 2.3.138: What can you say about two lines that have equal slopes and equal y...
 2.3.139: What can you say about two lines with the same xintercept and the ...
 2.3.140: If two distinct lines have the same slope, but different xintercep...
 2.3.141: If two distinct lines have the same yintercept, but different slop...
 2.3.142: Which form of the equation of a line do you prefer to use? Justify ...
 2.3.143: What Went Wrong? A student is asked to find the slope of the line j...
 2.3.144: Slope Open the slope applet. Move point B around the Cartesian plan...
Solutions for Chapter 2.3: Lines
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 2.3: Lines
Get Full SolutionsChapter 2.3: Lines includes 144 full stepbystep solutions. 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 144 problems in chapter 2.3: Lines have been answered, more than 8128 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321716811.

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

Column space C (A) =
space of all combinations of the columns of A.

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

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

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.

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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.

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

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).
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