 3.6.1: Give the equation that each picture models.
 3.6.2: Copy and fill in the table to solve the equation as in Example A.
 3.6.3: Give the next stages of the equation, matching the action taken, to...
 3.6.4: Complete the tables to solve the equations.
 3.6.5: Give the additive inverse of each number. a. b. 17 c. 23 d. x
 3.6.6: A multiplicative inverse is a number or expression that you can mul...
 3.6.7: Solve these equations. Tell what action you take at each stage. a. ...
 3.6.8: MiniInvestigation A solution to the equation 10 + 3x = 5 is shown ...
 3.6.9: Solve the equation 4 + 1.2x = 12.4 by using each method. a. balanci...
 3.6.10: Solve each equation symbolically using the balancing method. a. 3 +...
 3.6.11: You can solve familiar formulas for a specific variable. For exampl...
 3.6.12: An equation can have the variable on both sides. In these cases you...
 3.6.13: APPLICATION Economy drapes for a certain size window cost $90. They...
 3.6.14: Run the easy level of the LINES program on your calculator. [ See C...
 3.6.15: The local bagel store sells a bakers dozen of bagels for $6.49, whi...
Solutions for Chapter 3.6: Solving Equations Using the Balancing Method
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 3.6: Solving Equations Using the Balancing Method
Get Full SolutionsThis textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. Chapter 3.6: Solving Equations Using the Balancing Method includes 15 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. Since 15 problems in chapter 3.6: Solving Equations Using the Balancing Method have been answered, more than 5292 students have viewed full stepbystep solutions from this chapter.

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

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

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.

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

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)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·

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