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 2.1.2.1.1: The process of finding the number or numbers that make an equation ...
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 2.1.2.1.2: An equation in the form ax b c, such as 7x 9 13, is called a/an ___...
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 2.1.2.1.3: Equations that have the same solution are called ______________ equ...
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 2.1.2.1.4: The addition property of equality states that if a b, then a c ____...
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 2.1.2.1.6: The equation x 7 13 can be solved by ______________ to both sides.
 2.1.2.1.7: The equation 7x 5 6x can be solved by ______________ from both sides.
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 2.1.2.1.16: In Exercises 110, identify the linear equations in one variable. x 2 5
 2.1.2.1.17: In Exercises 110, identify the linear equations in one variable. x 5 8
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 2.1.2.1.62: The equations in Exercises 5558 contain small geometric figures tha...
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 2.1.2.1.66: In Exercises 5962, use the given information to write an equation. ...
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 2.1.2.1.70: Formulas frequently appear in the business world. For example, the ...
 2.1.2.1.71: Formulas frequently appear in the business world. For example, the ...
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 2.1.2.1.74: The diversity index, from 0 (no diversity) to 100, measures the cha...
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 2.1.2.1.76: State the addition property of equality and give an example.
 2.1.2.1.77: Explain why x 2 9 and x 2 6 are not equivalent equations.
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 2.1.2.1.79: Look, again, at the graph for Exercises 6768 that shows the diversi...
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 2.1.2.1.88: Write an equation with a negative solution that can be solved by ad...
 2.1.2.1.89: Use a calculator to solve each equation in Exercises 8283. x 7.0463...
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 2.1.2.1.91: Write as an algebraic expression in which x represents the number: ...
 2.1.2.1.92: Simplify: 16 8 4 (2). (Section 1.8, Example 4)
 2.1.2.1.93: Simplify: 37x 2(5x 1). (Section 1.8, Example 11)
 2.1.2.1.94: Exercises 8789 will help you prepare for the material covered in th...
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Solutions for Chapter 2.1: The Addition Property of Equality
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 2.1: The Addition Property of Equality
Get Full SolutionsThis textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. This expansive textbook survival guide covers the following chapters and their solutions. Since 101 problems in chapter 2.1: The Addition Property of Equality have been answered, more than 71122 students have viewed full stepbystep solutions from this chapter. Chapter 2.1: The Addition Property of Equality includes 101 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

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.

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.