 12.6.1: Graph the equation: y = 3x + 2
 12.6.2: Graph the equation: y + 4 = x2
 12.6.3: Graph the equation: y2 = x2  1
 12.6.4: Graph the equation: x 2 + 4y2 = 4
 12.6.5: In 524, graph each equation of the system.Then solve the system to ...
 12.6.6: In 524, graph each equation of the system.Then solve the system to ...
 12.6.7: In 524, graph each equation of the system.Then solve the system to ...
 12.6.8: In 524, graph each equation of the system.Then solve the system to ...
 12.6.9: In 524, graph each equation of the system.Then solve the system to ...
 12.6.10: In 524, graph each equation of the system.Then solve the system to ...
 12.6.11: In 524, graph each equation of the system.Then solve the system to ...
 12.6.12: In 524, graph each equation of the system.Then solve the system to ...
 12.6.13: In 524, graph each equation of the system.Then solve the system to ...
 12.6.14: In 524, graph each equation of the system.Then solve the system to ...
 12.6.15: In 524, graph each equation of the system.Then solve the system to ...
 12.6.16: In 524, graph each equation of the system.Then solve the system to ...
 12.6.17: In 524, graph each equation of the system.Then solve the system to ...
 12.6.18: In 524, graph each equation of the system.Then solve the system to ...
 12.6.19: In 524, graph each equation of the system.Then solve the system to ...
 12.6.20: In 524, graph each equation of the system.Then solve the system to ...
 12.6.21: In 524, graph each equation of the system.Then solve the system to ...
 12.6.22: In 524, graph each equation of the system.Then solve the system to ...
 12.6.23: In 2554,solve each system. Use any method you wish.
 12.6.24: In 2554,solve each system. Use any method you wish.
 12.6.25: In 2554,solve each system. Use any method you wish.
 12.6.26: In 2554,solve each system. Use any method you wish.
 12.6.27: In 2554,solve each system. Use any method you wish.
 12.6.28: In 2554,solve each system. Use any method you wish.
 12.6.29: In 2554,solve each system. Use any method you wish.
 12.6.30: In 2554,solve each system. Use any method you wish.
 12.6.31: In 2554,solve each system. Use any method you wish.
 12.6.32: In 2554,solve each system. Use any method you wish.
 12.6.33: In 2554,solve each system. Use any method you wish.
 12.6.34: In 2554,solve each system. Use any method you wish.
 12.6.35: In 2554,solve each system. Use any method you wish.
 12.6.36: In 2554,solve each system. Use any method you wish.
 12.6.37: In 2554,solve each system. Use any method you wish.
 12.6.38: In 2554,solve each system. Use any method you wish.
 12.6.39: In 2554,solve each system. Use any method you wish.
 12.6.40: In 2554,solve each system. Use any method you wish.
 12.6.41: In 2554,solve each system. Use any method you wish.
 12.6.42: In 2554,solve each system. Use any method you wish.
 12.6.43: In 2554,solve each system. Use any method you wish.
 12.6.44: In 2554,solve each system. Use any method you wish.
 12.6.45: In 2554,solve each system. Use any method you wish.
 12.6.46: In 2554,solve each system. Use any method you wish.
 12.6.47: In 2554,solve each system. Use any method you wish.
 12.6.48: In 2554,solve each system. Use any method you wish.
 12.6.49: In 2554,solve each system. Use any method you wish.
 12.6.50: In 2554,solve each system. Use any method you wish.
 12.6.51: In 2554,solve each system. Use any method you wish.
 12.6.52: In 2554,solve each system. Use any method you wish.
 12.6.53: In 2554,solve each system. Use any method you wish.
 12.6.54: In 2554,solve each system. Use any method you wish.
 12.6.55: Graph the equations given in Example 4.
 12.6.56: Graph the equations given in 49.
 12.6.57: In 5764, use a graphing utility to solve each system of equations. ...
 12.6.58: In 5764, use a graphing utility to solve each system of equations. ...
 12.6.59: In 5764, use a graphing utility to solve each system of equations. ...
 12.6.60: In 5764, use a graphing utility to solve each system of equations. ...
 12.6.61: In 5764, use a graphing utility to solve each system of equations. ...
 12.6.62: In 5764, use a graphing utility to solve each system of equations. ...
 12.6.63: In 5764, use a graphing utility to solve each system of equations. ...
 12.6.64: In 5764, use a graphing utility to solve each system of equations. ...
 12.6.65: In 6570, graph each equation and find the point(s) of intersection,...
 12.6.66: In 6570, graph each equation and find the point(s) of intersection,...
 12.6.67: In 6570, graph each equation and find the point(s) of intersection,...
 12.6.68: In 6570, graph each equation and find the point(s) of intersection,...
 12.6.69: In 6570, graph each equation and find the point(s) of intersection,...
 12.6.70: In 6570, graph each equation and find the point(s) of intersection,...
 12.6.71: The difference of two numbers is 2 and the sum of their squares is ...
 12.6.72: The sum of two numbers is 7 and the difference of their squares is ...
 12.6.73: The product of two numbers is 4 and the sum of their squares is 8. ...
 12.6.74: The product of two numbers is 10 and the difference of their square...
 12.6.75: The difference of two numbers is the same as their product, and the...
 12.6.76: The sum of two numbers is the same as their product, and the differ...
 12.6.77: The ratio of a to b is The sum of a and b is 10.What is the ratio o
 12.6.78: The ratio of a to b is The sum of a and b is 14. What is the ratio
 12.6.79: Geometry The perimeter of a rectangle is 16 inches and its area is ...
 12.6.80: Geometry An area of 52 square feet is to be enclosed by two squares...
 12.6.81: Geometry Two circles have circumferences that add up to centimeters...
 12.6.82: Geometry The altitude of an isosceles triangle drawn to its base is...
 12.6.83: The Tortoise and the Hare In a 21meter race between a tortoise and...
 12.6.84: Running a Race In a 1mile race,the winner crosses the finish line ...
 12.6.85: Constructing a Box A rectangular piece of cardboard, whose area is ...
 12.6.86: Constructing a Cylindrical Tube A rectangular piece of cardboard,wh...
 12.6.87: Fencing A farmer has 300 feet of fence available to enclose a 4500...
 12.6.88: Bending Wire A wire 60 feet long is cut into two pieces. Is it poss...
 12.6.89: Geometry Find formulas for the length l and width of a rectangle in...
 12.6.90: Geometry Find formulas for the base b and one of the equal sides l ...
 12.6.91: Descartess Method of Equal Roots Descartess method for finding tang...
 12.6.92: In 9298, use Descartess method from to find the equation of the lin...
 12.6.93: In 9298, use Descartess method from to find the equation of the lin...
 12.6.94: In 9298, use Descartess method from to find the equation of the lin...
 12.6.95: In 9298, use Descartess method from to find the equation of the lin...
 12.6.96: In 9298, use Descartess method from to find the equation of the lin...
 12.6.97: In 9298, use Descartess method from to find the equation of the lin...
 12.6.98: In 9298, use Descartess method from to find the equation of the lin...
 12.6.99: If and are two solutions of a quadratic equation it can be shown th...
 12.6.100: A circle and a line intersect at most twice. A circle and a parabol...
 12.6.101: Suppose that you are the manager of a sheet metal shop. A customer ...
Solutions for Chapter 12.6: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 12.6
Get Full SolutionsAlgebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 101 problems in chapter 12.6 have been answered, more than 57989 students have viewed full stepbystep solutions from this chapter. Chapter 12.6 includes 101 full stepbystep solutions.

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

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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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).

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

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.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).