 22.1: x2 + y2 = 4 2
 22.2: h(x) = 1.1  2x
 22.3: On that day, what was the value in U.S. dollars of 200 euros?
 22.4: On that day, what was the value in euros of 500 U.S. dollars?
 22.5: y = 3x  5
 22.6: 4x = 10y + 6 7
 22.7: y = _2 3 x + 1
 22.8: y = 3x  5
 22.9: x  y  2 = 0
 22.10: x + y = 5
 22.11: f(x) = 6x  19
 22.12: f(x) = 7x5 + x  1
 22.13: h(x) = 2x3  4x2 + 5
 22.14: g(x) = 10 + _2 x2
 22.15: 1 x + 3y = 5
 22.16: x + y = 4 1
 22.17: y = 2x  5 PH
 22.18: How far does a sound travel underwater in 5 seconds?
 22.19: In air, the equation is y = 343x. Does sound travel faster in air o...
 22.20: Find the temperature at a height of 10,000 feet.
 22.21: Find the height if the temperature is 58F.
 22.22: y = 3x + 4 2
 22.23: y = 12x
 22.24: x = 4y  5
 22.25: x = 7y + 2 2
 22.26: 5y = 10x  25 2
 22.27: 4x = 8y  12
 22.28: 5x + 3y = 15 2
 22.29: 2x  6y = 12 3
 22.30: 3x  4y  10 = 0
 22.31: 2x + 5y  10 = 0 3
 22.32: y = x
 22.33: y = 4x  2
 22.34: GEOMETRY Find the area of the shaded region in the graph. (Hint: Th...
 22.35: 1 2 x + _1 2 y = 6
 22.36: 1 3 x  _1 3 y = 2 3
 22.37: 0.5x = 3
 22.38: 0.25y = 10
 22.39: 5 6 x + _1 15y = _3 10
 22.40: 0.25x = 0.1 + 0.2y
 22.41: y = 2
 22.42: y = 4 4
 22.43: x = 8 4
 22.44: 3x + 2y = 6
 22.45: x = 1
 22.46: f(x) = 4x  1 4
 22.47: g(x) = 0.5x  3 4
 22.48: 4x + 8y = 12
 22.49: ATMOSPHERE Graph the linear function in Exercises 20 and 21.
 22.50: Write an equation that is a model for the different numbers of maga...
 22.51: Graph the equation. Does this equation represent a function? Explain.
 22.52: If Latonya sells 100 magazine subscriptions and 200 newspaper subsc...
 22.53: OPEN ENDED Write an equation of a line with an xintercept of 2.
 22.54: REASONING Explain why f(x) = _x + 2 2 is a linear function.
 22.55: Graph the equations. Then compare and contrast the graphs.
 22.56: Write a linear equation whose graph is between the graphs of x + y ...
 22.57: REASONING Explain why the graph of x + 3y = 0 has only one intercept.
 22.58: Writing in Math Use the information about study time on page 66 to ...
 22.59: ACT/SAT Which function is linear? A f(x) = x2 B g(x) = 2.7 C f(x) =...
 22.60: REVIEW What is the complete solution to the equation? 9  3x = 18 F...
 22.61: {(1, 5), (1, 3), (2, 4), (4, 3)}
 22.62: {(0, 2), (1, 3), (2, 1), (1, 0)}
 22.63: 2 < 3x + 1 < 7
 22.64: x + 4 > 2
 22.65: TAX Including a 6% sales tax, a paperback book costs $8.43. What is...
 22.66: 4
 22.67: 1 2
 22.68: 3 _3 4
 22.69: 1.25
Solutions for Chapter 22: Linear Equations
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 22: Linear Equations
Get Full SolutionsSince 69 problems in chapter 22: Linear Equations have been answered, more than 60735 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Chapter 22: Linear Equations includes 69 full stepbystep solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

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).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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