 32.1: y = 3x  4 y = 4 + x
 32.2: 4c + 2d = 10 c + 3d = 10
 32.3: a  b = 2 4. 3g  2h = 1 2a + 3b = 3
 32.4: 3g  2h = 1 2a + 3b = 3 4g + h = 17
 32.5: STANDARDIZED TEST PRACTICE Campus Rentals rents 2 and 3bedroom ap...
 32.6: 2r  3s = 11 7. 5m + n = 10 8. 2p + 4q = 18 2r + 2s = 6
 32.7: 5m + n = 10 8. 2p + 4q = 18 2r + 2s = 6 4m + n = 4
 32.8: 2p + 4q = 18 2r + 2s = 6 4m + n = 4 3p  6q = 3
 32.9: _1 4 x + y = _11 4 10. _1 6 y  2 = _1 9 11. 1.25x  y = 7 x  _1 ...
 32.10: 1 6 y  2 = _1 9 11. 1.25x  y = 7 x  _1 2 y = 2 12 = 18y
 32.11: 1.25x  y = 7 x  _1 2 y = 2 12 = 18y 4y = 5x + 28
 32.12: 2j 3k = 3 13. 2r + s = 11 14. 5a b = 17 j + k = 14
 32.13: 2r + s = 11 14. 5a b = 17 j + k = 14 6r  2s = 2
 32.14: 5a b = 17 j + k = 14 6r  2s = 2 3a + 2b = 5
 32.15: w  z = 2 16. 3s + 2t = 3 17. 2x + 4y = 6 4w + 5z = 16
 32.16: 3s + 2t = 3 17. 2x + 4y = 6 4w + 5z = 16 s + _1 3 t = 4
 32.17: 2x + 4y = 6 4w + 5z = 16 s + _1 3 t = 4 7x = 4 + 3y
 32.18: u + v = 7 19. m n = 9 20. r + 4s = 8 2u + v = 11
 32.19: m n = 9 20. r + 4s = 8 2u + v = 11 7m + n = 7
 32.20: r + 4s = 8 2u + v = 11 7m + n = 7 3r + 2s = 6
 32.21: 4x  5y = 17 22. 2c + 6d = 14 23. 6d + 3f = 12 3x + 4y =
 32.22: 2c + 6d = 14 23. 6d + 3f = 12 3x + 4y = 5  _ 7 3 + _1 3 c = d
 32.23: 6d + 3f = 12 3x + 4y = 5  _ 7 3 + _1 3 c = d 2d = 8  f
 32.24: Write a system of equations that represents the number of members w...
 32.25: How many members rented skis and how many rented snowboards?
 32.26: Write a system of two equations that represents the number of each ...
 32.27: How many laser printers and how many color monitors were delivered?
 32.28: 3p  6q = 6 29. 10m  9n = 15 30. 3c  7d = 3 2p  4q = 4
 32.29: 10m  9n = 15 30. 3c  7d = 3 2p  4q = 4 5m  4n = 10
 32.30: 3c  7d = 3 2p  4q = 4 5m  4n = 10 2c + 6d = 34
 32.31: 6g  8h = 50 32. 2p = 7 + q 33. 3x = 31 + 2y 6h = 22  4g
 32.32: 2p = 7 + q 33. 3x = 31 + 2y 6h = 22  4g 6p  3q = 24
 32.33: 3x = 31 + 2y 6h = 22  4g 6p  3q = 24 5x + 6y = 23
 32.34: 3u + 5v = 6 35. 3a = 3 + 2b 36. 0.25x + 1.75y = 1.25 2u 4v = 7
 32.35: 3a = 3 + 2b 36. 0.25x + 1.75y = 1.25 2u 4v = 7 3a + b = 3
 32.36: 0.25x + 1.75y = 1.25 2u 4v = 7 3a + b = 3 0.5x + 2 = 2.5y
 32.37: 8 = 0.4m + 1.8n 38. s + 3t = 27 39. 2f + 2g = 18 1.2m + 3.4n = 16
 32.38: s + 3t = 27 39. 2f + 2g = 18 1.2m + 3.4n = 16 2t = 19  _1 2 s
 32.39: 2f + 2g = 18 1.2m + 3.4n = 16 2t = 19  _1 2 s _1 6 f + _1 3 g = 1
 32.40: Write a system of equations that represents the number of each type...
 32.41: How many of each type of question will be on the test?
 32.42: If most of his students can answer true/false questions within 1 mi...
 32.43: Write a system of equations that represents Megans morning workout.
 32.44: How long should she do each activity in order to burn 335 Calories?
 32.45: OPEN ENDED Give a system of equations that is more easily solved by...
 32.46: REASONING Make a conjecture about the solution of a system of equat...
 32.47: FIND THE ERROR Juanita and Jamal are solving the system 2x y = 6 an...
 32.48: CHALLENGE Solve the system of equations. _1 x + _3 y = _3 4 _3 x  ...
 32.49: Writing in Math Use the information on page 123 to explain how a sy...
 32.50: ACT/SAT In order to practice at home, Tadeo purchased a basketball ...
 32.51: REVIEW The caterer at a brunch bought several pounds of chicken sal...
 32.52: y = x + 2 53. 4y  2x = 4 54. 3x + y = 1 y = x  1
 32.53: 4y  2x = 4 54. 3x + y = 1 y = x  1 y  _1 2 x = 1
 32.54: 3x + y = 1 y = x  1 y  _1 2 x = 1 y = 2x  4
 32.55: x + y 3
 32.56: 5y  4x < 20
 32.57: 3x + 9y 15
 32.58: y = 7x + 4
 32.59: x = y
 32.60: 3x = 2  5y
 32.61: 6x = 3y  9
 32.62: y = _1 2 x  3
 32.63: 2 3 y  6 = 1  x
 32.64: ELECTRICITY Find the amount of current I (in amperes) produced if t...
 32.65: 3x + 2y 10; (2, 1)
 32.66: 4x  2y > 6; (3, 3)
 32.67: 7x + 4y 15; (4, 2)
Solutions for Chapter 32: Solving Systems of Equations Algebraically
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 32: Solving Systems of Equations Algebraically
Get Full SolutionsChapter 32: Solving Systems of Equations Algebraically includes 67 full stepbystep solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This expansive textbook survival guide covers the following chapters and their solutions. Since 67 problems in chapter 32: Solving Systems of Equations Algebraically have been answered, more than 56034 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1.

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

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column space C (A) =
space of all combinations of the columns of A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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 .

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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