 1.1.1.1.8: In Exercises 710, find the slope of the line passing through the pa...
 1.1.1.1.9: In Exercises 710, find the slope of the line passing through the pa...
 1.1.1.1.10: In Exercises 710, find the slope of the line passing through the pa...
 1.1.1.1.11: In Exercises 1118, use the point on the line and the slope of the l...
 1.1.1.1.12: In Exercises 1118, use the point on the line and the slope of the l...
 1.1.1.1.13: In Exercises 1118, use the point on the line and the slope of the l...
 1.1.1.1.14: In Exercises 1118, use the point on the line and the slope of the l...
 1.1.1.1.15: In Exercises 1118, use the point on the line and the slope of the l...
 1.1.1.1.16: In Exercises 1118, use the point on the line and the slope of the l...
 1.1.1.1.17: In Exercises 1118, use the point on the line and the slope of the l...
 1.1.1.1.18: In Exercises 1118, use the point on the line and the slope of the l...
 1.1.1.1.19: In Exercises 1924, (a) find the slope and yintercept (if possible)...
 1.1.1.1.20: In Exercises 1924, (a) find the slope and yintercept (if possible)...
 1.1.1.1.21: In Exercises 1924, (a) find the slope and yintercept (if possible)...
 1.1.1.1.22: In Exercises 1924, (a) find the slope and yintercept (if possible)...
 1.1.1.1.23: In Exercises 1924, (a) find the slope and yintercept (if possible)...
 1.1.1.1.24: In Exercises 1924, (a) find the slope and yintercept (if possible)...
 1.1.1.1.25: In Exercises 2532, find the general form of the equation of the lin...
 1.1.1.1.26: In Exercises 2532, find the general form of the equation of the lin...
 1.1.1.1.27: In Exercises 2532, find the general form of the equation of the lin...
 1.1.1.1.28: In Exercises 2532, find the general form of the equation of the lin...
 1.1.1.1.29: In Exercises 2532, find the general form of the equation of the lin...
 1.1.1.1.30: In Exercises 2532, find the general form of the equation of the lin...
 1.1.1.1.31: In Exercises 2532, find the general form of the equation of the lin...
 1.1.1.1.32: In Exercises 2532, find the general form of the equation of the lin...
 1.1.1.1.33: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.34: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.35: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.36: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.37: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.38: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.39: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.40: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.41: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.42: In Exercises 33 42, find the slopeintercept form of the equation o...
 1.1.1.1.43: In Exercises 43 and 44, find the slopeintercept form of the equati...
 1.1.1.1.44: In Exercises 43 and 44, find the slopeintercept form of the equati...
 1.1.1.1.45: Annual Salary A jewelers salary was $28,500 in 2004 and $32,900 in ...
 1.1.1.1.46: Annual Salary A librarians salary was $25,000 in 2004 and $27,500 i...
 1.1.1.1.47: In Exercises 4750, determine the slope and yintercept of the linea...
 1.1.1.1.48: In Exercises 4750, determine the slope and yintercept of the linea...
 1.1.1.1.49: In Exercises 4750, determine the slope and yintercept of the linea...
 1.1.1.1.50: In Exercises 4750, determine the slope and yintercept of the linea...
 1.1.1.1.51: In Exercises 51 and 52, use a graphing utility to graph the equatio...
 1.1.1.1.52: In Exercises 51 and 52, use a graphing utility to graph the equatio...
 1.1.1.1.53: In Exercises 5356, determine whether the lines and passing through ...
 1.1.1.1.54: In Exercises 5356, determine whether the lines and passing through ...
 1.1.1.1.55: In Exercises 5356, determine whether the lines and passing through ...
 1.1.1.1.56: In Exercises 5356, determine whether the lines and passing through ...
 1.1.1.1.57: In Exercises 57 62, write the slopeintercept forms of the equation...
 1.1.1.1.58: In Exercises 57 62, write the slopeintercept forms of the equation...
 1.1.1.1.59: In Exercises 57 62, write the slopeintercept forms of the equation...
 1.1.1.1.60: In Exercises 57 62, write the slopeintercept forms of the equation...
 1.1.1.1.61: In Exercises 57 62, write the slopeintercept forms of the equation...
 1.1.1.1.62: In Exercises 57 62, write the slopeintercept forms of the equation...
 1.1.1.1.63: In Exercises 63 and 64, the lines are parallel. Find the slopeinter...
 1.1.1.1.64: In Exercises 63 and 64, the lines are parallel. Find the slopeinter...
 1.1.1.1.65: In Exercises 65 and 66, the lines are perpendicular. Find the slope...
 1.1.1.1.66: In Exercises 65 and 66, the lines are perpendicular. Find the slope...
 1.1.1.1.67: Graphical Analysis In Exercises 6770, identify any relationships th...
 1.1.1.1.68: Graphical Analysis In Exercises 6770, identify any relationships th...
 1.1.1.1.69: Graphical Analysis In Exercises 6770, identify any relationships th...
 1.1.1.1.70: Graphical Analysis In Exercises 6770, identify any relationships th...
 1.1.1.1.71: Earnings per Share The graph shows the earnings per share of stock ...
 1.1.1.1.72: Sales The graph shows the sales (in billions of dollars) for Goodye...
 1.1.1.1.73: Height The rise to run ratio of the roof of a house determines the ...
 1.1.1.1.74: Road Grade When driving down a mountain road, you notice warning si...
 1.1.1.1.75: Rate of Change In Exercises 7578, you are given the dollar value of...
 1.1.1.1.76: Rate of Change In Exercises 7578, you are given the dollar value of...
 1.1.1.1.77: Rate of Change In Exercises 7578, you are given the dollar value of...
 1.1.1.1.78: Rate of Change In Exercises 7578, you are given the dollar value of...
 1.1.1.1.79: Graphical Interpretation In Exercises 79 82, match the description ...
 1.1.1.1.80: Graphical Interpretation In Exercises 79 82, match the description ...
 1.1.1.1.81: Graphical Interpretation In Exercises 79 82, match the description ...
 1.1.1.1.82: Graphical Interpretation In Exercises 79 82, match the description ...
 1.1.1.1.83: Depreciation A school district purchases a highvolume printer, cop...
 1.1.1.1.84: Meteorology Recall that water freezes at and boils at (a) Find an e...
 1.1.1.1.85: Cost, Revenue, and Profit A contractor purchases a bulldozer for $3...
 1.1.1.1.86: Rental Demand A real estate office handles an apartment complex wit...
 1.1.1.1.87: Education In 1991, Penn State University had an enrollment of 75,34...
 1.1.1.1.88: Writing Using the results of Exercise 87, write a short paragraph d...
 1.1.1.1.89: Synthesis True or False? In Exercises 89 and 90, determine whether ...
 1.1.1.1.90: Synthesis True or False? In Exercises 89 and 90, determine whether ...
 1.1.1.1.91: Exploration In Exercises 9194, use a graphing utility to graph the ...
 1.1.1.1.92: Exploration In Exercises 9194, use a graphing utility to graph the ...
 1.1.1.1.93: Exploration In Exercises 9194, use a graphing utility to graph the ...
 1.1.1.1.94: Exploration In Exercises 9194, use a graphing utility to graph the ...
 1.1.1.1.95: In Exercises 9598, use the results of Exercises 9194 to write an eq...
 1.1.1.1.96: In Exercises 9598, use the results of Exercises 9194 to write an eq...
 1.1.1.1.97: In Exercises 9598, use the results of Exercises 9194 to write an eq...
 1.1.1.1.98: In Exercises 9598, use the results of Exercises 9194 to write an eq...
 1.1.1.1.99: Library of Parent Functions In Exercises 99 and 100, determine whic...
 1.1.1.1.100: Library of Parent Functions In Exercises 99 and 100, determine whic...
 1.1.1.1.101: Library of Parent Functions In Exercises 101 and 102, determine whi...
 1.1.1.1.102: Library of Parent Functions In Exercises 101 and 102, determine whi...
 1.1.1.1.103: Think About It Does every line have both an xintercept and a yint...
 1.1.1.1.104: Think About It Can every line be written in slopeintercept form? E...
 1.1.1.1.105: Think About It Does every line have an infinite number of lines tha...
 1.1.1.1.106: Think About It Does every line have an infinite number of lines tha...
 1.1.1.1.107: In Exercises 107112, determine whether the expression is a polynomi...
 1.1.1.1.108: In Exercises 107112, determine whether the expression is a polynomi...
 1.1.1.1.109: In Exercises 107112, determine whether the expression is a polynomi...
 1.1.1.1.110: In Exercises 107112, determine whether the expression is a polynomi...
 1.1.1.1.111: In Exercises 107112, determine whether the expression is a polynomi...
 1.1.1.1.112: In Exercises 107112, determine whether the expression is a polynomi...
 1.1.1.1.113: In Exercises 113116, factor the trinomial.
 1.1.1.1.114: In Exercises 113116, factor the trinomial.
 1.1.1.1.115: In Exercises 113116, factor the trinomial.
 1.1.1.1.116: In Exercises 113116, factor the trinomial.
 1.1.1.1.117: Make a Decision To work an extended application analyzing the numbe...
Solutions for Chapter 1.1: Lines in the Plane
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 1.1: Lines in the Plane
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Since 110 problems in chapter 1.1: Lines in the Plane have been answered, more than 48220 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Chapter 1.1: Lines in the Plane includes 110 full stepbystep solutions.

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

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).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

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.

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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 •

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.