 28.1: Solve each equation or formula for the variable specified.
 28.2: Solve each equation or formula for the variable specified.
 28.3: Solve each equation or formula for the variable specified.
 28.4: Solve each equation or formula for the variable specified.
 28.5: GEOMETRY For Exercises 5 and 6, use the formula for the area of a t...
 28.6: GEOMETRY For Exercises 5 and 6, use the formula for the area of a t...
 28.7: SWIMMING A swimmer swims about one third of a lap per minute. At th...
 28.8: Solve each equation or formula for the variable specified.
 28.9: Solve each equation or formula for the variable specified.
 28.10: Solve each equation or formula for the variable specified.
 28.11: Solve each equation or formula for the variable specified.
 28.12: Solve each equation or formula for the variable specified.
 28.13: Solve each equation or formula for the variable specified.
 28.14: Solve each equation or formula for the variable specified.
 28.15: Solve each equation or formula for the variable specified.
 28.16: Solve each equation or formula for the variable specified.
 28.17: Solve each equation or formula for the variable specified.
 28.18: Solve each equation or formula for the variable specified.
 28.19: Solve each equation or formula for the variable specified.
 28.20: Solve each equation or formula for the variable specified.
 28.21: Solve each equation or formula for the variable specified.
 28.22: GEOMETRY For Exercises 22 and 23, use the formula for the area of a...
 28.23: GEOMETRY For Exercises 22 and 23, use the formula for the area of a...
 28.24: WORK For Exercises 24 and 25, use the following information. The fo...
 28.25: WORK For Exercises 24 and 25, use the following information. The fo...
 28.26: Solve each equation or formula for the variable specified.
 28.27: Solve each equation or formula for the variable specified.
 28.28: Solve each equation or formula for the variable specified.
 28.29: Write an equation and solve for the variable specified.
 28.30: Write an equation and solve for the variable specified.
 28.31: Write an equation and solve for the variable specified.
 28.32: DANCING For Exercises 32 and 33, use the following information. The...
 28.33: DANCING For Exercises 32 and 33, use the following information. The...
 28.34: PACKAGING The Yummy Ice Cream Company wants to package ice cream in...
 28.35: CHALLENGE Write a formula for the area of the arrow. Describe how y...
 28.36: REASONING Describe the possible values of t if s = _r t  2 . Expla...
 28.37: OPEN ENDED Write a formula for A, the area of a geometric figure su...
 28.38: Writing in Math Use the information on page 117 to explain how equa...
 28.39: If 2x + y = 5, what is the value of 4x? A 10  y B 10  2y C _ 5  ...
 28.40: REVIEW What is the base of the triangle if the area is 56 meters sq...
 28.41: FOOD In order for a food to be marked reduced fat, it must have at ...
 28.42: Solve each proportion.
 28.43: Solve each proportion.
 28.44: Solve each proportion.
 28.45: PREREQUISITE SKILL Use the Distributive Property to rewrite each ex...
 28.46: PREREQUISITE SKILL Use the Distributive Property to rewrite each ex...
 28.47: PREREQUISITE SKILL Use the Distributive Property to rewrite each ex...
 28.48: PREREQUISITE SKILL Use the Distributive Property to rewrite each ex...
Solutions for Chapter 28: Solving for a Specific Variable
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 28: Solving for a Specific Variable
Get Full SolutionsAlgebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. Since 48 problems in chapter 28: Solving for a Specific Variable have been answered, more than 35570 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 28: Solving for a Specific Variable includes 48 full stepbystep solutions.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

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

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

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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.

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.

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·

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