 4.1.4.1.1: Determine whether each given ordered pair is a solution of each sys...
 4.1.4.1.2: Determine whether each given ordered pair is a solution of each sys...
 4.1.4.1.3: Determine whether each given ordered pair is a solution of each sys...
 4.1.4.1.4: Determine whether each given ordered pair is a solution of each sys...
 4.1.4.1.5: Determine whether each given ordered pair is a solution of each sys...
 4.1.4.1.6: Determine whether each given ordered pair is a solution of each sys...
 4.1.4.1.7: Determine whether each given ordered pair is a solution of each sys...
 4.1.4.1.8: Determine whether each given ordered pair is a solution of each sys...
 4.1.4.1.9: Solve each system by graphing. See Examples 2, through 4.e x + y = ...
 4.1.4.1.10: Solve each system by graphing. See Examples 2, through 4.e2x  y = ...
 4.1.4.1.11: Solve each system by graphing. See Examples 2, through 4. e2y  4x ...
 4.1.4.1.12: Solve each system by graphing. See Examples 2, through 4.4x  y = 6...
 4.1.4.1.13: Solve each system by graphing. See Examples 2, through 4.e3x  y = ...
 4.1.4.1.14: Solve each system by graphing. See Examples 2, through 4. e x + 3y...
 4.1.4.1.15: Solve each system of equations by the substitution method. See Exam...
 4.1.4.1.16: Solve each system of equations by the substitution method. See Exam...
 4.1.4.1.17: Solve each system of equations by the substitution method. See Exam...
 4.1.4.1.18: Solve each system of equations by the substitution method. See Exam...
 4.1.4.1.19: Solve each system of equations by the substitution method. See Exam...
 4.1.4.1.20: Solve each system of equations by the substitution method. See Exam...
 4.1.4.1.21: Solve each system of equations by the substitution method. See Exam...
 4.1.4.1.22: Solve each system of equations by the substitution method. See Exam...
 4.1.4.1.23: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.24: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.25: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.26: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.27: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.28: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.29: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.30: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.31: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.32: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.33: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.34: Solve each system of equations by the elimination method. See Examp...
 4.1.4.1.35: Solve each system of equations.e2x + 5y = 86x + y = 10
 4.1.4.1.36: Solve each system of equations.e x  4y = 53x  8y = 0
 4.1.4.1.37: Solve each system of equations. e2x + 3y = 1x  2y = 4
 4.1.4.1.38: Solve each system of equations.e 2x + y = 8x + 3y = 11
 4.1.4.1.39: Solve each system of equations.13x + y = 43 14x  12y =  14
 4.1.4.1.40: Solve each system of equations.34x  12y =  12x + y =  32
 4.1.4.1.41: Solve each system of equations.e2x + 6y = 83x + 9y = 12
 4.1.4.1.42: Solve each system of equations.e x = 3y  12x  6y = 2
 4.1.4.1.43: Solve each system of equations.e4x + 2y = 52x + y = 1
 4.1.4.1.44: Solve each system of equations.e3x + 6y = 152x + 4y = 3
 4.1.4.1.45: Solve each system of equations.e10y  2x = 15y = 4  6x
 4.1.4.1.46: Solve each system of equations.e3x + 4y = 07x = 3y
 4.1.4.1.47: Solve each system of equations.e 5x  2y = 273x + 5y = 18
 4.1.4.1.48: Solve each system of equations.e3x + 4y = 22x + 5y = 1
 4.1.4.1.49: Solve each system of equations.e x = 3y + 25x  15y = 10
 4.1.4.1.50: Solve each system of equations.y = 17x + 3x  7y = 21
 4.1.4.1.51: Solve each system of equations.e2x  y = 1y = 2x
 4.1.4.1.52: Solve each system of equations.x = 15yx  y = 4
 4.1.4.1.53: Solve each system of equations.e2x = 6y = 5  x
 4.1.4.1.54: Solve each system of equations.e x = 3y + 4y = 5
 4.1.4.1.55: Solve each system of equations.x + 52 = 6  4y33x5 = 21  7y10
 4.1.4.1.56: Solve each system of equations.y5 = 8  x2x = 2y  83
 4.1.4.1.57: Solve each system of equations.e 4x  7y = 712x  21y = 24
 4.1.4.1.58: Solve each system of equations.e 2x  5y = 124x + 10y = 20
 4.1.4.1.59: Solve each system of equations.23x  34y = 1 16x +38y = 1
 4.1.4.1.60: Solve each system of equations.12x  13y = 318x +16y = 0
 4.1.4.1.61: Solve each system of equations.e0.7x  0.2y = 1.60.2x  y = 1.4
 4.1.4.1.62: Solve each system of equations.e0.7x + 0.6y = 1.30.5x  0.3y = 0.8
 4.1.4.1.63: Solve each system of equations.e4x  1.5y = 10.22x + 7.8y = 25.68
 4.1.4.1.64: Solve each system of equations.e x  3y = 5.36.3x + 6y = 3.96
 4.1.4.1.65: Determine whether the given replacement values make each equation t...
 4.1.4.1.66: Determine whether the given replacement values make each equation t...
 4.1.4.1.67: Determine whether the given replacement values make each equation t...
 4.1.4.1.68: Determine whether the given replacement values make each equation t...
 4.1.4.1.69: Add the equations. See Section 4.1.3x + 2y  5z = 103x + 4y + z = 15
 4.1.4.1.70: Add the equations. See Section 4.1.x + 4y  5z = 202x  4y  2z = 17
 4.1.4.1.71: Add the equations. See Section 4.1.10x + 5y + 6z = 149x + 5y  6z ...
 4.1.4.1.72: Add the equations. See Section 4.1.9x  8y  z = 319x + 4y  z = 12
 4.1.4.1.73: Without graphing, determine whether each system has one solution, n...
 4.1.4.1.74: Without graphing, determine whether each system has one solution, n...
 4.1.4.1.75: Without graphing, determine whether each system has one solution, n...
 4.1.4.1.76: Without graphing, determine whether each system has one solution, n...
 4.1.4.1.77: Can a system consisting of two linear equations have exactly two so...
 4.1.4.1.78: Suppose the graph of the equations in a system of two equations in ...
 4.1.4.1.79: The concept of supply and demand is used often in business. In gene...
 4.1.4.1.80: The concept of supply and demand is used often in business. In gene...
 4.1.4.1.81: The concept of supply and demand is used often in business. In gene...
 4.1.4.1.82: The concept of supply and demand is used often in business. In gene...
 4.1.4.1.83: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.84: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.85: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.86: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.87: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.88: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.89: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.90: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.91: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.92: The revenue equation for a certain brand of toothpaste is y = 2.5x,...
 4.1.4.1.93: Solve each system. To do so, you may want to let a = 1 x (if x is i...
 4.1.4.1.94: Solve each system. To do so, you may want to let a = 1 x (if x is i...
 4.1.4.1.95: Solve each system. To do so, you may want to let a = 1 x (if x is i...
 4.1.4.1.96: Solve each system. To do so, you may want to let a = 1 x (if x is i...
 4.1.4.1.97: Solve each system. To do so, you may want to let a = 1 x (if x is i...
 4.1.4.1.98: Solve each system. To do so, you may want to let a = 1 x (if x is i...
 4.1.4.1.99: Solve each system. To do so, you may want to let a = 1 x (if x is i...
 4.1.4.1.100: Solve each system. To do so, you may want to let a = 1 x (if x is i...
Solutions for Chapter 4.1: Solving Systems of Linear Equations in Two Variables
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 4.1: Solving Systems of Linear Equations in Two Variables
Get Full SolutionsSince 100 problems in chapter 4.1: Solving Systems of Linear Equations in Two Variables have been answered, more than 66277 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.1: Solving Systems of Linear Equations in Two Variables includes 100 full stepbystep solutions.

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.

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

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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

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.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.