 23.1: Solve each equation. Check your solution. _t 7 = 5
 23.2: Solve each equation. Check your solution. _a 36 = _4 9
 23.3: Solve each equation. Check your solution. _2 3 n = 10
 23.4: Solve each equation. Check your solution. _8 9 = _4 5 k
 23.5: Solve each equation. Check your solution. 12 = _x 3
 23.6: Solve each equation. Check your solution.  _r 4 = _1 7
 23.7: GEOGRAPHY The discharge of a river equals the product of its width,...
 23.8: Solve each equation. Check your solution. 8t = 72
 23.9: Solve each equation. Check your solution. 20 = 4w
 23.10: Solve each equation. Check your solution. 45 = 9a
 23.11: Solve each equation. Check your solution. 2g = 84
 23.12: Write an equation for each problem. Then solve the equation. Five t...
 23.13: Write an equation for each problem. Then solve the equation. One th...
 23.14: Solve each equation. Check your solution. _x 9 = 10
 23.15: Solve each equation. Check your solution. _b 7 = 11
 23.16: Solve each equation. Check your solution. _3 4 = _c 24
 23.17: Solve each equation. Check your solution. _2 3 = _1 8 y
 23.18: Solve each equation. Check your solution. _2 3 n = 14
 23.19: Solve each equation. Check your solution. _3 5 g = 6
 23.20: Solve each equation. Check your solution. 4 _1 5 = 3p
 23.21: Solve each equation. Check your solution. 5 = 3 _1 2 x
 23.22: Solve each equation. Check your solution. 6 =  _1 2 n
 23.23: Solve each equation. Check your solution.  _2 5 = _z 45
 23.24: Solve each equation. Check your solution.  _ g 24 = _5 12
 23.25: Solve each equation. Check your solution.  _v 5 = 45
 23.26: GENETICS About two twentyfifths of the male population in the worl...
 23.27: WORLD RECORDS In 1998, Winchells House of Donuts in Pasadena, Calif...
 23.28: Solve each equation. Check your solution. 8d = 48
 23.29: Solve each equation. Check your solution. 65 = 13t
 23.30: Solve each equation. Check your solution. 5r = 55
 23.31: Solve each equation. Check your solution. 252 = 36s
 23.32: Solve each equation. Check your solution. 58 = 29h
 23.33: Solve each equation. Check your solution. 26a = 364
 23.34: Write an equation for each problem. Then solve the equation. Seven ...
 23.35: Write an equation for each problem. Then solve the equation. Two fi...
 23.36: Write an equation for each problem. Then solve the equation. Negati...
 23.37: Write an equation for each problem. Then solve the equation. Twelve...
 23.38: Solve each equation. Check your solution. (3 _14)p = 2 _1 2
 23.39: Solve each equation. Check your solution. 5h = 3 _2 3
 23.40: Solve each equation. Check your solution. (2 _3 5)t = 22
 23.41: Solve each equation. Check your solution. 3.15 = 1.5y
 23.42: Solve each equation. Check your solution. 11.78 = 1.9f
 23.43: Solve each equation. Check your solution. 2.8m = 9.8
 23.44: Write an equation for each problem. Then solve the equation. Negati...
 23.45: Write an equation for each problem. Then solve the equation. Two an...
 23.46: Write an equation for each problem. Then solve the equation. One an...
 23.47: BASEBALL For Exercises 47 and 48, use the following information. In...
 23.48: BASEBALL For Exercises 47 and 48, use the following information. In...
 23.49: PHYSICAL SCIENCE For Exercises 4951, use the following information....
 23.50: PHYSICAL SCIENCE For Exercises 4951, use the following information....
 23.51: PHYSICAL SCIENCE For Exercises 4951, use the following information....
 23.52: OPEN ENDED Write a multiplication equation that has a solution of ...
 23.53: REASONING Compare and contrast the Multiplication Property of Equal...
 23.54: CHALLENGE Discuss how you can use 6y  7 = 4 to find the value of 1...
 23.55: FIND THE ERROR Casey and Camila are solving 8n = 72. Who is correc...
 23.56: Writing in Math Use the data about light speed on page 85 to explai...
 23.57: REVIEW Which is the best estimate for the number of minutes on the ...
 23.58: REVIEW Mr. Morisson is draining his cylindrical, above ground pool....
 23.59: Solve each equation. Check your solution. m + 14 = 81
 23.60: Solve each equation. Check your solution. d  27 = 14
 23.61: Solve each equation. Check your solution. 17  (w) = 55
 23.62: Translate the following sentence into an equation. (Lesson 21) Ten...
 23.63: MUSIC Ryan practiced playing his violin 40 minutes on Monday and n ...
 23.64: PREREQUISITE SKILL Use the order of operations to find each value. ...
 23.65: PREREQUISITE SKILL Use the order of operations to find each value. ...
 23.66: PREREQUISITE SKILL Use the order of operations to find each value. ...
 23.67: PREREQUISITE SKILL Use the order of operations to find each value. ...
Solutions for Chapter 23: Solving Equations by Using Multiplication and Division
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 23: Solving Equations by Using Multiplication and Division
Get Full SolutionsAlgebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. Chapter 23: Solving Equations by Using Multiplication and Division includes 67 full stepbystep solutions. Since 67 problems in chapter 23: Solving Equations by Using Multiplication and Division have been answered, more than 35182 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.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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

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

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.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.