 2.2.1.821: In Exercises 17, solve each equation. 4x 5 13
 2.2.1.822: In Exercises 17, solve each equation. 12x 4 7x 21
 2.2.1.823: In Exercises 17, solve each equation. 8 5(x 2) x 26
 2.2.1.824: In Exercises 17, solve each equation. 3(2y 4) 9 3(y 1)
 2.2.1.825: In Exercises 17, solve each equation. 3 4 x 15
 2.2.1.826: In Exercises 17, solve each equation. x 10 1 3 x 5 1 2
 2.2.1.827: In Exercises 17, solve each equation. 9.2x 80.1 21.3x 19.6
 2.2.1.828: The formula P 2.4x 180 P, in millions x years after 1960. How many ...
 2.2.1.829: In Exercises 910, solve each formula for the specified variable. V ...
 2.2.1.830: In Exercises 910, solve each formula for the specified variable. l ...
 2.2.1.831: What is 6% of 140?
 2.2.1.832: 120 is 80% of what?
 2.2.1.833: 12 is what percent of 240? In Exercises 1418, solve each problem.
 2.2.1.834: In Exercises 1418, solve each problem. The product of 5 and a numbe...
 2.2.1.835: In Exercises 1418, solve each problem. Compared with other major co...
 2.2.1.836: In Exercises 1418, solve each problem. A phone plan has a monthly f...
 2.2.1.837: In Exercises 1418, solve each problem. A rectangular field is twice...
 2.2.1.838: In Exercises 1418, solve each problem. After a 20% reduction, you p...
 2.2.1.839: In Exercises 1921, find the area of each figure.
 2.2.1.840: In Exercises 1921, find the area of each figure.
 2.2.1.841: In Exercises 1921, find the area of each figure.
 2.2.1.842: In Exercises 2223, find the volume of each figure. Whereapplicable,...
 2.2.1.843: In Exercises 2223, find the volume of each figure. Whereapplicable,...
 2.2.1.844: What will it cost to cover a rectangular floor measuring 40 feet by...
 2.2.1.845: A sailboat has a triangular sail with an area of 56 square feet and...
 2.2.1.846: In a triangle, the measure of the first angle is three times that o...
 2.2.1.847: How many degrees are there in an angle that measures 16 more than t...
 2.2.1.848: x 2
 2.2.1.849: x 3
 2.2.1.850: x 2 3
 2.2.1.851: 6 9x 33
 2.2.1.852: 4x 2 2(x 6)
 2.2.1.853: A student has grades on three examinations of 76, 80, and 72. What ...
 2.2.1.854: The length of a rectangle is 20 inches. For what widths is the peri...
Solutions for Chapter 2: Chapter 2 Test
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 2: Chapter 2 Test
Get Full SolutionsIntroductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. Since 34 problems in chapter 2: Chapter 2 Test have been answered, more than 74816 students have viewed full stepbystep solutions from this chapter. Chapter 2: Chapter 2 Test includes 34 full stepbystep solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Outer product uv T
= column times row = rank one matrix.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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}.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.