 25.1: Solve each equation. Check your solution.
 25.2: Solve each equation. Check your solution.
 25.3: Solve each equation. Check your solution.
 25.4: Solve each equation. Check your solution.
 25.5: NUMBER THEORY Four times the greater of two consecutive integers is...
 25.6: Solve each equation. Check your solution.
 25.7: Solve each equation. Check your solution.
 25.8: Solve each equation. Check your solution.
 25.9: Solve each equation. Check your solution.
 25.10: STANDARDIZED TEST EXAMPLE Find the value of x so that the figures h...
 25.11: Solve each equation. Check your solution.
 25.12: Solve each equation. Check your solution.
 25.13: Solve each equation. Check your solution.
 25.14: Solve each equation. Check your solution.
 25.15: Solve each equation. Check your solution.
 25.16: Solve each equation. Check your solution.
 25.17: Solve each equation. Check your solution.
 25.18: Solve each equation. Check your solution.
 25.19: Solve each equation. Check your solution.
 25.20: Solve each equation. Check your solution.
 25.21: Solve each equation. Check your solution.
 25.22: Solve each equation. Check your solution.
 25.23: Solve each equation. Check your solution.
 25.24: Solve each equation. Check your solution.
 25.25: One half of a number increased by 16 is four less than two thirds o...
 25.26: The sum of one half of a number and 6 equals one third of the numbe...
 25.27: Two less than one third of a number equals 3 more than one fourth o...
 25.28: Two times a number plus 6 is three less than one fifth of the numbe...
 25.29: NUMBER THEORY Twice the greater of two consecutive odd integers is ...
 25.30: NUMBER THEORY Three times the greatest of three consecutive even in...
 25.31: Solve each equation. Check your solution.
 25.32: Solve each equation. Check your solution.
 25.33: Solve each equation. Check your solution.
 25.34: Solve each equation. Check your solution.
 25.35: HEALTH When exercising, a persons pulse rate should not exceed a ce...
 25.36: HARDWARE Traditionally, nails are given names such as 2penny, 3pe...
 25.37: Solve each equation. Check your solution.
 25.38: Solve each equation. Check your solution.
 25.39: Solve each equation. Check your solution.
 25.40: Solve each equation. Check your solution.
 25.41: Solve each equation. Check your solution.
 25.42: Solve each equation. Check your solution.
 25.43: ANALYZE TABLES The table shows the households that had Brand A and ...
 25.44: GEOMETRY The rectangle and square shown at right have the same peri...
 25.45: CHALLENGE Write an equation that has one or more grouping symbols, ...
 25.46: OPEN ENDED Find a counterexample to the statement All equations hav...
 25.47: REASONING Determine whether each solution is correct. If the soluti...
 25.48: Writing in Math Use the information about Internet users on page 98...
 25.49: Which equation represents the second step of the solution process? ...
 25.50: REVIEW Tanya sells cosmetics doortodoor. She makes $5 an hour and ...
 25.51: Solve each equation. Check your solution.
 25.52: Solve each equation. Check your solution.
 25.53: Solve each equation. Check your solution.
 25.54: HEALTH A female burns 4.5 Calories per minute pushing a lawn mower....
 25.55: A teacher took a survey of his students to find out how many televi...
 25.56: Write an algebraic expression for each verbal expression. Then simp...
 25.57: Write an algebraic expression for each verbal expression. Then simp...
 25.58: Find the solution set for each inequality, given the replacement set.
 25.59: Find the solution set for each inequality, given the replacement set.
 25.60: Evaluate each expression when a = 5, b = 8, and c = 1.
 25.61: Evaluate each expression when a = 5, b = 8, and c = 1.
 25.62: Evaluate each expression when a = 5, b = 8, and c = 1.
 25.63: PREREQUISITE SKILL Simplify each fraction.
 25.64: PREREQUISITE SKILL Simplify each fraction.
 25.65: PREREQUISITE SKILL Simplify each fraction.
 25.66: PREREQUISITE SKILL Simplify each fraction.
Solutions for Chapter 25: Solving Equations with the Variable on Each Side
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 25: Solving Equations with the Variable on Each Side
Get Full SolutionsChapter 25: Solving Equations with the Variable on Each Side includes 66 full stepbystep solutions. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. This expansive textbook survival guide covers the following chapters and their solutions. Since 66 problems in chapter 25: Solving Equations with the Variable on Each Side have been answered, more than 34043 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.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

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

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

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·