 P2.1: In Exercises 13, simplify the expression. 14x2y3 32x1y
 P2.2: In Exercises 13, simplify the expression. 860 2135 15
 P2.3: In Exercises 13, simplify the expression. 28x4y3
 P2.4: In Exercises 4 6, perform the operation and simplify the result. 4x...
 P2.5: In Exercises 4 6, perform the operation and simplify the result. . ...
 P2.6: In Exercises 4 6, perform the operation and simplify the result. 2 ...
 P2.7: In Exercises 79, factor the expression completely. 36 (x 4)2
 P2.8: In Exercises 79, factor the expression completely. x 5x2 6x
 P2.9: In Exercises 79, factor the expression completely. 54 16x3
 P2.10: Find the midpoint of the line segment connecting the points (7 2, 4...
 P2.11: Write the standard form of the equation of a circle with center (1 ...
 P2.12: In Exercises 1214, use point plotting to sketch the graph of the eq...
 P2.13: In Exercises 1214, use point plotting to sketch the graph of the eq...
 P2.14: In Exercises 1214, use point plotting to sketch the graph of the eq...
 P2.15: In Exercises 1517, (a) write the general form of the equation of th...
 P2.16: In Exercises 1517, (a) write the general form of the equation of th...
 P2.17: In Exercises 1517, (a) write the general form of the equation of th...
 P2.18: Find the equation of the line that passes through the point (2, 3) ...
 P2.19: In Exercises 19 and 20, evaluate the function at each specified val...
 P2.20: In Exercises 19 and 20, evaluate the function at each specified val...
 P2.21: In Exercises 2124, find the domain of the function. f(x) = (x + 2)(...
 P2.22: In Exercises 2124, find the domain of the function. . f(t) = 5 + 7t
 P2.23: In Exercises 2124, find the domain of the function. g(s) = 9 s2
 P2.24: In Exercises 2124, find the domain of the function. h(x) = 4 5x + 2
 P2.25: Determine whether the function given by g(x) = 3x x3 is even, odd, ...
 P2.26: Does the graph at the right represent y as a function of x? Explain.
 P2.27: Use a graphing utility to graph the function f(x) = 2x 5 x + 5. The...
 P2.28: Compare the graph of each function with the graph of f(x) = x. (a) ...
 P2.29: In Exercises 2932, evaluate the indicated function for f(x) = x2 + ...
 P2.30: In Exercises 2932, evaluate the indicated function for f(x) = x2 + ...
 P2.31: In Exercises 2932, evaluate the indicated function for f(x) = x2 + ...
 P2.32: In Exercises 2932, evaluate the indicated function for f(x) = x2 + ...
 P2.33: In Exercises 3335, determine whether the function has an inverse fu...
 P2.34: In Exercises 3335, determine whether the function has an inverse fu...
 P2.35: In Exercises 3335, determine whether the function has an inverse fu...
 P2.36: In Exercises 3639, solve the equation algebraically. Then write the...
 P2.37: In Exercises 3639, solve the equation algebraically. Then write the...
 P2.38: In Exercises 3639, solve the equation algebraically. Then write the...
 P2.39: In Exercises 3639, solve the equation algebraically. Then write the...
 P2.40: In Exercises 40 43, solve the inequality and sketch the solution on...
 P2.41: In Exercises 40 43, solve the inequality and sketch the solution on...
 P2.42: In Exercises 40 43, solve the inequality and sketch the solution on...
 P2.43: In Exercises 40 43, solve the inequality and sketch the solution on...
 P2.44: A soccer ball has a volume of about 370.7 cubic inches. Find the ra...
 P2.45: A rectangular plot of land with a perimeter of 546 feet has a width...
 P2.46: The table shows the net profits P (in millions of dollars) for McDo...
Solutions for Chapter P2: Solving Equations and Inequalities
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter P2: Solving Equations and Inequalities
Get Full SolutionsSince 46 problems in chapter P2: Solving Equations and Inequalities have been answered, more than 61040 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Chapter P2: Solving Equations and Inequalities includes 46 full stepbystep solutions.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

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].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

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.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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).

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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

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.